Theory of Reproducing Kernels

Theory of Reproducing KernelsAuthor(s): N. AronszajnSource: Transactions of the American Mathematical Society, Vol. 68, No. 3 (May, 1950), pp.337-404Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/1990404 .

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THEORY OF REPRODUCING KERNELS(1) BY

N. ARONSZAJN

PREFACE

The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie gen6rale de noyaux reproduisants-Premiere partie (vol. 39 (1944)) which was written in 1942-1943. In the introduction to this paper we outlined the plan of papers which were to follow. In the meantime, however, the general theory has been developed in many directions, and our original plans have had to be changed.

Due to wartime conditions we were not able, at the time of writing the first paper, to take into account all the earlier investigations which, although sometimes of quite a different character, were, nevertheless, related to our subject.

Our investigation is concerned with kernels of a special type which have been used under different names and in different ways in many domains of mathematical research. We shall therefore begin our present paper with a short historical introduction in which we shall attempt to indicate the different manners in which these kernels have been used by various investigators, and to clarify the terminology. We shall also discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter.

In Part I, we shall discuss briefly the essential notions and results of our previous paper and give a further development of the theory in an abstract form. In Part II, we shall illustrate the results obtained in the first part by a series of examples which will give new developments of already known applications of the theory, as well as some new applications.

TABLE OF CONTENTS

HISTORICAL INTRODUCTION ...................................................... 338 PART I. GENERAL THEORY .......................................... . . 342

1. Definition of reproducing kernels ........................................... 342 2. Resum6 of basic properties of reproducing kernels ............................ 343 3. Reproducing kernels of finite-dimensional classes ....... ...................... 346 4. Completion of incomplete Hilbert spaces ............ ........................ 347 5. The restriction of a reproducing kernel ...................................... 350 6. Sum of reproducing kernels ................................................ 352

Presented to the Society, December 28, 1948, under the title Hilbert spaces and conformal mappings; received by the editors November 26, 1948.

(1) Paper done under contract with the Office of Naval Research, N5ori 76-16-NR 043-046, Division of Engineering Science, Harvard University.

337



338 N. ARONSZAJN [May

7. Difference of reproducing kernels ............... ............................ 354 8. Product of reproducing kernels ............. ................................ 357 9. Limits of reproducing kernels .............................................. 362

10. Construction of a r.k. by resolution of identity .... ........................ 368 11. Operators in spaces with reproducing kernels ................................ 371 12. The reproducing kernel of a sum of two closed subspaces ..................... 375 13. Final remarks in the general theory ........................................ 380

PART II. EXAMPLES ............................................................ 384 1. Introductory remarks ..................................................... 384

(1) Bergman's kernels .................................................... 384 (2) Harmonic kernels ..................................................... 386

2. Comparison domains ..................................................... 387 3. The difference of kernels .................................................. 388 4. The square of a kernel introduced by Szegb ................................. 391 5. The kernel H(z, zi) for an ellipse ... ....................................... 393 6. Construction of H(z, z1) for a strip .......................................... 394 7. Limits of increasing sequences of kernels .................................... 396 8. Construction of reproducing kernels by the projection-formula of ?12, I ......... 397

BIBLIOGRAPHY .......................... .................................. . 401

HISTORICAL INTRODUCTION

Examples of kernels of the type in which we are interested have been known for a long time, since all the Green's functions of self-adjoint ordinary differential equations (as also some Green's functions-the bounded ones- of partial differential equations) belong to this type. But the characteristic properties of these kernels as we now understand them have only been stressed and applied since the beginning of the century.

There have been and continue to be two trends in the consideration of these kernels. To explain them we should mention that such a kernel K(x, y) may be characterized as a function of two points, by a property discovered by J. Mercer [1 ](2) in 1909. To the kernel K there corresponds a well determined class F of functions f(x), in respect to which K possesses the "reproducing" property (E. H. Moore [2]). On the other hand, to a class of functions F, there may correspond a kernel K with "reproducing" property (N. Aronszajn [4 ]).

Those following the first trend consider a given kernel K and study it in itself, or eventually apply it in various domains (as integral equations, theory of groups, general metric geometry, and so on). The class F corresponding to K may be used as a tool of research, but is introduced a posteriori (as in the work of E. H. Moore [2], and more recently of A. Weil [1], I. Gelfand and D. Raikoff [1], and R. Godement [1, 2]). In the second trend, one is interested primarily in a class of functions F, and the corresponding kernel K is used essentially as a tool in the study of the functions of this class. One of the basic problems in this kind of investigation is the explicit construction and computation of the kernel for a given class F.

(2) Numbers in brackets refer to the bibliography at the end of the paper.



19501 THEORY OF REPRODUCING KERNELS 339

The first of these trends originated in the theory of integral equations as developed by Hilbert. The kernels considered then were continuous kernels of positive definite integral operators. This theory was developed by J. Mercer [1, 2] under the name of "positive definite kernels" and on occasion has been used by many others interested in integral equations, especially during the second decade of this century. Mercer discovered the property

n

(1) E K(yi, yj)&ijj _ 0, yi any points, ti any complex numbers('),

characterizing his kernels, among all the continuous kernels of integral equations. To this same trend belong the investigations of E. H. Moore [1, 2] who, during the second and third decades of the century, introduced these kernels in the general analysis under the name of "positive hermitian matrices" with a view to applications in a kind of generalization of integral equations. Moore considered kernels K(x, y) defined on an abstract set E and characterized by the property (1). He discovered the theorem now serving as one of the links between the two trends, proving that to each positive hermitian matrix there corresponds a class of functions forming what we now call a Hilbert space with a scalar product (f, g) and in which the kernel has the reproducing property.

(2) f(y) = (f(x), K(x, y)).

Also to the same trend (though seemingly without any connection to previous investigations) belongs the notion introduced by S. Bochner [2] during the third decade of the century under the name of "positive definite functions." Bochner considered continuous functions +(x) of real variable x such that the kernels K(x, y) =q$(x-y) conformed to condition (1). He introduced these functions with a view to application in the theory of Fourier transforms. The notion was later generalized by A. Weil [1 ] and applied by I. Gelfand and D. Raikoff [1], R. Godement [1, 2], and others to the investigation of topological groups under the name of positive definite functions or functions of positive type. These functions were also applied to general metric geometry (the Hilbert distances) by I. J. Schoenberg [1, 2], J. v. Neumann and I. J. Schoenberg [1], and S. Bochner [3].

The second trend was initiated during the first decade of the century in the work of S. Zaremba [1, 2] on boundary value problems for harmonic and biharmonic functions. Zaremba was the first to introduce, in a particular case, the kernel corresponding to a class of functions, and to state its reproducing property (2). However, he did not develop any general theory, nor did he give any particular name to the kernels he introduced. It appears that nothing was done in this direction until the third decade when S. Berg-

(3) Mercer used only real numbers (j as he considered only real kernels K.




man [1] introduced kernels corresponding to classes of harmonic functions and analytic functions in one or several variables. He called these "kernel functions." They were introduced as kernels of orthogonal systems in these classes for an adequate metric. The reproducing property of these kernels was noticed by Bergman [1] (also by N. Aronszajn [1]), but it was not used as their basic characteristic property as is done at present.

In the third and fourth decades most of the work was done with kernels which we shall call Bergman's kernels, that is, kernels of classes of analytic functionsf of one or several complex variables, regular in a domain D with the quadratic metric

ID f12dr.

A quantity of important results were achieved by the use of these kernels in the theory of functions of one and several complex variables (Bergman [4, 6, 7], Bochner [1]), in conformal mapping of simply- and multiply-connected domains (Bergman [1 1, 12 ], Zarankiewicz [1 ] and others), in pseudo- conformal mappings (Bergman [4, 5, 8, 9], Welke [1], Aronszajn [1], and others), in the study of invariant Riemannian metrics (Bergman [11, 14], Fuchs [1, 2]), and in other subjects.

The original idea of Zaremba to apply the kernels to the solution of boundary value problems was represented in these two decades by only a few papers of Bergman [1, 2, 3, 10]. Only since the last war has this idea been put into the foreground by a series of papers by Bergman [13], and Bergman and Schiffer [1, 2, 3]. In these investigations, the kernel was proved to be a powerful tool for solving boundary value problems of partial differential equations of elliptic type. By the use of variational methods going back to Hada- mard, relations were established between the kernels corresponding to classes of solutions of different equations and for different domains (Bergman and Schiffer [1, 3]). For a partial differential equation, the kernel of the class of solutions in a domain was proved to be the difference of the corresponding Neumann's and Green's functions (Bergman and Schiffer [1, 3]) (in the special case of the biharmonic equation a relation of this kind was already noticed by Zaremba). Parallel to this revival of the application of kernels to partial differential equations there is developing a study of the relationship between these kernels and Bergman's kernels of analytic functions (Bergman [12 ], M. Schiffer [1, 2 ]). Also the application of kernels to conformal mapping of multiply-connected domains has made great progress as all the important mapping functions were proved to be simply expressible by the Bergman's kernel (Bergman [I1, 12], P. Garabedian and M. Schiffer [1], Garabedian [1], and Z. Nehari [1, 2]). Quite recently, the connection was found between the Bergman's kernel and the kernel introduced by G. Szego (P. Garabedian [1 ]).

In 1943, the author ([4] also [6]) developed the general theory of repro-



1950] THEORY OF REPRODUCING KERNELS 341

ducing kernels which contains, as particular cases, the Bergman kernelfunctions. This theory gives a general basis for the study of each particular case and allows great simplification of many of the proofs involved. In this theory a central role is played by the reproducing property of the kernel in respect to the class to which it belongs. The kernel is defined by this property. The simple fact was stressed that a reproducing kernel always possesses property (1) characteristic of positive hermitian matrices (in the sense of E. H. Moore). This forms the second link between the two trends in the kernel theory (the theorem of E. H. Moore forming the first link was mentioned above) .

The mathematicians working in the two trends seem not to have noticed the essential connections between the general notions they were using. At present the two concepts of the kernel, as a positive hermitian matrix and as a reproducing kernel, are known to be equivalent and methods elaborated in the investigations belonging to one trend prove to be of importance in the other.

We should like to elaborate here briefly on the matter of the terminology which has been used by various investigators. As we have seen above, different names have been given to the kernels in which we are interested. When the kernels were used in themselves, without special or previous consideration of the class to which they belonged, they were called "positive definite kernels," "positive hermitian matrices," "positive definite functions," or "functions of positive type." In cases where they were considered as determined, and in connection with a class of functions, they were called "kernel functions" or "reproducing kernels." It is not our intention to settle here the question of terminology. Our purpose is rather to state our choice and to give our reasons for it together with a comparison of the terminology we have chosen with that used by other authors.

It would seem advisable to keep two names for our kernels, the function of each name being to indicate immediately in what context the kernel under consideration is to be taken. Thus, when we consider the kernel in itself we shall call it (after E. H. Moore) a positive matrix(4), in abbreviation, p. matrix, or p. m. When we wish to indicate the kernel corresponding to a class of functions we shall call it the reproducing kernel of the class, in abbreviation, r. kernel or r.k.

As compared to other terminology, we believe that the name "positive definite function" or perhaps better "function of positive type" will probably continue to be used in the particular case when the kernel is of the form +(x-y), x, y belonging to an additive group. This term has been used in a few instances for some more general kernels, but we believe that it would prove to be more convenient if it were restricted to the particular case

(4) We drop here the adjective "hermitian" since the condition that the quadratic form (1) be positive implies the hermitian symmetry of the matrix.




mentioned above. Although the name of "positive definite kernels" would seem, somehow,

more adequate than "positive matrices," especially since it was introduced first, we have chosen rather the term used by E. H. Moore. This is because we wish to reserve the notion of positive definite kernel for more general kernels which would include the positive definite matrices as well as some other non-bounded kernels (such as the kernels of general positive definite integral operators and also the recently introduced pseudo-reproducing kernels [Aronszajn 5, 6]).

On the other hand, when we have in mind the kernel corresponding to a given class of functions, the simplest terminology is to call it "the kernel of the class" or "the kernel belonging to the class." But when some ambiguity is to be feared, or when we wish to stress its characteristic property, we use the adjective "reproducing."

PART I. GENERAL THEORY

1. Definition of reproducing kernels. Consider a linear class F of functions f(x) defined in a set E. We shall suppose that F is a complex class, that is, that it admits of multiplication by complex constants.

Suppose further that forfECFis defined a norm |fI (that is, a real number satisfying: | fH |-, | f||=? only for f=O, |cf| =|cj |fl 1If+gI fI-g<IfI -+| gI|) given by a quadratic hermitian form Q(f)

HlJfl2 = Q(). Here, a functional Q(f) is called quadratic hermitian if for any constants

41, 02 and functions fi, f2 of F

Q(1ifl + 42f2) = I 41 12Q(fl) + S1&2Q(fb, f2) + &12Q(f2, fl) + 1 t2 12Q(f2).

Q(fl, f2) = Q(f2, fi) is the uniquely determined bilinear hermitian form corresponding to the quadratic form Q(f). This bilinear form will be denoted by (fi, f2) Q(fi, f2) and called the scalar product corresponding to the norm

(or the quadratic metric IIf 12). We have

H 2= (ff) The class F with the norm, ||, forms a normed complex vector space.

If this space is complete it is a Hilbert space. If F is a class of real-valued functions forming a real vector space (that is,

admitting of multiplication with only real constants), if the norm, || ||, in F is given by I|S||2 = Q(f) with an ordinary quadratic form Q (that is, for real 01, 42, Q(Ilf+02f2) = Q(fl) + 2i12Q(fl, f2)+2 Q (p, where Q(fl, f2) is the corresponding bilinear symmetric form), and if F is a complete space, it is a real Hilbert space. The scalar product is given there by (fi, f2) Q(fi,f2).

Every class F of real functions forming a real Hilbert space determines a




complex Hilbert space in the following way: consider all functions fl+if2 with fi and f2 in F. They form a complex vector space F, in which we define the norm by L|f1+if2H|2=H|fi112+l1f2H12. F, is a complex Hilbert space.

The complex Hilbert spaces F determined in this way by real Hilbert spaces are characterized by the two properties:

(1) if fEF, fEF (f is the conjugate complex function of f), (2) IfII =Y. Let F be a class of functions defined in E, forming a Hilbert space (complex

or real). The function K(x, y) of x and y in E is called a reproducing kernel (r.k.) of F if

1. For every y, K(x, y) as function of x belongs to F. 2. The reproducing property: for every y CE and every fE F,

f(y) = (f(x), K(x, y))x.

The subscript x by the scalar product indicates that the scalar product applies to functions of x.

If a real class F possesses a r.k. K(x, y) then it is immediately verified that the corresponding complex space F, possesses the same kernel (which is real- valued):

From now on (unless otherwise stated) we shall consider only complex Hilbert spaces. As we have seen there is no essential limitation in this assumption.

It will be useful to introduce a distinction between the terms subclass and subspace. When F, and F2 are two classes of functions defined in the same set E, F, is a subclass of F2 if everyf of F, belongs to F2. F, is a subspace of F2 if it is a subclass of F2 and if for every fEF,, IfII 1= fI 2 (|| ||1 and || 112 are the norms in F, and F2 respectively). F,CF2 means that F, is a subclass of F2.

2. Resume of basic properties of reproducing kernels. In the following, F denotes a class of functions f(x) defined in E, forming a Hilbert space with the norm 11fjf and scalar product (f1,f2). K(x, y) will denote the corresponding reproducing kernel.

The detailed proofs of the properties listed below may be found in [Aronszajn, 4].

(1) If a r.k. K exists it is unique. In fact, if another K'(x, y) existed we would have for some y

O < K(x, y) - K'(x, Y)H2 = (K - K', K - K') = (K - K', K) - (K - K', K') = 0

because of the reproducing property of K and K'. (2) Existence. For the existence of a r.k. K(x, y) it is necessary and suffi-

cient that for every y of the set E, f(y) be a continuous functional of f running through the Hilbert space F.

In fact, if K exists, then |f(y) I _ ?|fl| (K(x, y), K(x, y)) 112 = K(y, y)1IflfI| I.




On the other hand if f(y) is a continuous functional, then by the general theory of the Hilbert space there exists a function gy(x) belonging to F such that f(y) = (f(x), gy(x)), and then if we put K(x, y) -g(x) it will be a reproducing kernel.

(3) K(x, y) is a positive matrix in the sense of E. H. Moore, that is, the quadratic form in 01, * , in

n

(1) Z K(yi, yj)%itj i ,j=l

is non-negative for all yl, , yn in E. This is clear since expression (1) equals | K(x, yi) 2, following the reproducing property. In particular it follows that

K(x, x) > 0, K(x, y) = K(y, x), |K(X ) 2 < K(x, x) K(y, y).

(4) The theorem in (3) admits a converse due essentially to E. H. Moore: to every positive matrix K(x, y) there corresponds one and only one class of functions with a uniquely determined quadratic form in it, forming a Hilbert space and admitting K(x, y) as a reproducing kernel.

This class of functions is generated by all the functions of the form ZEakK(x, y). The norm of this function is defined by the quadratic form || Ek K(x, yk)| 2 =E EK(yi, yj) ijj. Functions with this norm do not as yet form a complete Hilbert space, but it can be easily seen that they may be completed by the adjunction of functions to form such a complete Hilbert space. This follows from the fact that every Cauchy sequence of these functions (relative to the above norm) will converge at every point x towards a limit function whose adjunction to the class will complete the space.

(5) If the class F possesses a r.k. K(x, y), every sequence of functions {fn } which converges strongly to a functionf in the Hilbert space F, converges also at every point in the ordinary sense, lim fn(x) =f(x), this convergence being uniform in every subset of E in which K(x, x) is uniformly bounded. This follows from

| f(y) - fn(y) = (f(x) - fn(x), K(x, y)) K -K(x, y)

- f-fnll(K(y, y))112

If fn converges weakly to f, we have again fn(y) ->f(y) for every y (since, by the definition of the weak convergence, (fn(x), K(x, y))->(f(x), K(x, y)). There is in general no increasing sequence of sets EjCE2C ->E in each of which fn converges uniformly to f.

If a topology (a notion of limit) is defined in E and if the correspondence y<-+K(x, y) transforms E in a continuous manner into a subset of the space F, then the weakly convergent sequence {f, } converges uniformly in every compact set E1CE. In fact E1 is transformed into a compact subset of the




space F. For every E >0 we can then choose a finite set (yl, , yj) CE1 so that for every y El there exists at least one Yk with I|K(x, y)-K(x, y1)fl <E/4M, where M=I.u.b., IfA||

Further, if we choose no so that for n>no, If(yk) -f.(yk)I <e/4, we shall obtain for yCE1.

| f(y) - f.(y) | = (f(y) - f(yk)) + (f(yk) - fk(yk)) + (fA(yk) - fn(Y)) I

I f(y7) - fn(y1) + I (f(x) -- f.(x), K(x, y) - K(x, y)) |

< -+ If - fJI IIK(x, y) - K(x, y&)II < - + 2M i- < e. 4 =4 4M1

The continuity of the correspondence y*-*K(x, y) is equivalent to equi- continuity of all functions of F with Ilf I < M for any M> 0. This property is satisfied by most of the classes with reproducing kernels which are usually considered (such as classes of analytic functions, harmonic functions, solutions of partial differential equations, and so on).

(6) If the class F witlh the r.k. K is a subspace of a larger Hilbert space .-, then the formula

f(y) (It, K(x, y))x

gives the projection f of the element h of S on F. In fact h =f+g, where g is orthogonal to the class F. K(x, y) as a function

of x belongs to F and so we have (h, K(x, y)) - (f+g, K(x, y))=(f, K(x, y)) =f(y) by the reproducing property.

(7) If F possesses a r.k. K, then the same is true of all closed linear subspaces of F, because if f(y) is a continuous functional of f running through F, it is so much the more so if f runs through a subclass of F. If F' and F" are complementary subspaces of F, then their reproducing kernels satisfy the equation K'+ -K" = K.

(8) If F possesses a r.k. K and if { g. } is an orthonormal system in F, then for every sequence { ax } of numbers satisfying

00

ZIanI2< %}

we have

Ia I IXl gn(X) I _ K(x, x)1/2 E 2)

In fact, for a fixed y, the Fourier coefficients of K(x, y) for the system { gn } are

(K(x, y), g.(x)) = (g.(x), K(.x, y)) = gn(Y)




Consequently 00 > gn (y)2 ? (K (x, y), K(x, y)) x = K(y, y). 1

Therefore oo 00O 1/ / 2 oo \ 12

an gn(X) an 2 gn(X) 12)

< K(x, x)11/2 jEan12)

3. Reproducing kernels of finite-dimensional classes. If F is of finite dimension n, let wi(x), * * *, wn(x) be n linearly independent functions of F. All functions f(x) of F are representable in a unique manner as

(1) f(x) = > rkWk(X), k complex constants.

The most general quadratic metric in F will be given by a positive definite hermitian form

(2) 2 a= Ej7 ?ii i, j=l

The scalar product has the form

(3) (f, g) = a ijj, where g =E -OkWk. i,i

It is clear that

(4) a= (wi, w;).

Therefore the matrix { aij} is the Gramm's matrix of the system {Wkw. This matrix always possesses an inverse. Denote by {f3ij} the conjugate of this inverse matrix. We have then

(5) E aci4jk = 0 or 1 following as i # k or i = k.

It is immediately verified that the function

(6) K(x, y) = E 3j jwj(x)wj(y) i, i=1

is the reproducing kernel of the class F with the metric (2). The matrix {13ij} is hermitian positive definite. From the preceding de-

velopments we get, clearly, the following theorem.




THEOREM. A function K(x, y) is the reproducing kernel of a finite-dimensional class of functions if and only if it is of the form (6) with a positive definite matrix {I ij } and linearly independent functions wk(x). The corresponding class F is then generated by the functions Wk(X), the functions f EF given by (1) and the corresponding norm IIf I given by (2), where { aij } is the inverse matrix of

4. Completion of incomplete Hilbert spaces. In applications, we often meet classes of functions forming incomplete Hilbert spaces, that is, linear classes, with a scalar product, satisfying all the conditions for a Hilbert space with the exception of the completeness. For such classes, two problems present themselves. Firstly, the problem of completing the class so as to obtain a class of functions forming a complete Hilbert space and secondly, to decide (before effecting the completion of the class) if the complete class will possess a reproducing kernel.

A few remarks should be added here about the problem of the completion of a class of functions forming an incomplete Hilbert space. Consider such a class F. It is well known that to this class we can adjoin ideal elements which will be considered as the limits of Cauchy sequences in F, when such a limit is not available in F, and in such a way we obtain an abstract Hilbert space containing the class F as a dense subset. This space, however, will not form a class of functions. In quite an arbitrary way we could realize the ideal elements to be adjoined to F as functions so as to obtain a complete space formed by a linear class of functions, but, in general, this arbitrary manner of completion will destroy all the continuity properties between the values of the functions and the convergence in the space.

In this paper when we speak about the functional completion of an incomplete class of functions F, we mean a completion by adjunction of functions such that the value of a function f of the completed class at a given point y depends continuously on f (as belonging to the Hilbert space) (5) . From the existence theorem of reproducing kernels we deduce the fact that a completed class has a reproducing kernel. In this way the problem of functional completion and of the existence of a reproducing kernel in the complete class is merged into one problem. We shall prove here the following theorem:

THEOREM. Consider a class of functions F forming an incomplete Hilbert space. In order that there exist a functional completion of the class it is necessary and sufficient that 10 for every fixed y LE the linear functional f(y) defined in F be bounded; 20 for a Cauchy sequence {fm } CF, the condition fm(y) ->O for every y implies I|fm! ->O. If the functional completion is possible, it is unique.

Proof. That the first condition is necessary is immediately seen from the

(5) A more general functional completion was introduced in connection with the theory of pseudo-reproducing kernels (N. Aronszajn [5 ]).




existence theorem of reproducing kernels, since the complete class would necessarily have such a kernel. The necessity of the second condition follows from the fact that a Cauchy sequence in F is strongly convergent in the complete space to a function f, and the function f is the limit of fm at every point y of E. Consequently, f = 0 and the norms of fm have to converge to the norm of f which is equal to zero. To prove the sufficiency we proceed as follows: consider any Cauchy sequence {fn}CF. For every fixed y denote by 1U, the bound of the functional f(y) so that

(1) ~~~~~~~f(Y) I -< M^,||fil.

Consequently,

I fm(y) - fn(y) ? _ My| fm - n|| .

It follows that {fn(y) } is a Cauchy sequence of complex numbers, that is, it converges to a number which we shall denote by f(y). In this way the Cauchy sequence {fn } defines a function f to which it is convergent at every point of E.

Consider the class of all the functions f, limits of Cauchy sequences {fn} CF. It is immediately seen that it is a linear class of functions, and that it contains the class F (since the Cauchy sequence {f,, } with fn, =f C F is obviously convergent tof). Consider, then, in the so-defined class F, the norm

(2) lf Ill = lim llf,ll

for any Cauchy sequence {f,,} CF converging to f at every point y. This norm does not depend on the choice of the Cauchy sequence: in fact, if another sequence, {f"' } converges to f at every point y, then fn' -f, will be a Cauchy sequence converging to zero, and by the second condition the norms fr'-fnjj converge to zero.

Consequently,

r limflfj - lrn | = lim r - |fn < limn flf, f 0.

On the other hand, it is readily seen that IIf112 is a quadratic positive form in the class F; it is obviously 0 for f= 0, and it is positive for f 0 because of (1). This norm defines a scalar product in F satisfying all the required properties. It remains to be shown that F is complete and contains F as a dense subspace.

The second assertion is immediately proved because FC F. For elements of F,the norms l |l| 1|| flcoincide,and everyfunctionfCFis, bydefinition, the limit of a Cauchy sequence {fn } CF everywhere in E. It follows that f is a strong limit of fn in F since by (2)

lim I If - f'f I lim lim r lfm - |II = 0. nf=O n-wo m-=o




To prove the first assertion, that is, the completeness of F, we shall consider any Cauchy sequence {f,, } CTF. Since F is dense in F, we can find a Cauchy sequence Cf~ } F such that

limlfnf - fn[i = 0.

The Cauchy sequence {f,n' } converges to a f unction fCG T. This convergence is meant at first as ordinary convergence everywhere in E, but the argument used above shows that the f7,' also converge strongly to f in the space F. It follows immediately that fn converges strongly to f. The uniqueness of the complete class is seen from the fact that iLn the completed class a function.f must necessarily be a strong limit of a Cauchy sequence {fn. } CF. Since a reproducing kernel must exist for the completed class, this implies that f is a limit everywhere of the Cauchy sequence If, } which means that it belongs to the above class F7. As the norm of f has to be the limit of 1ff41 it necessarily coincides with Ijf IJ1. It is also clear that every function f of F must belong to the completed class. In summing up the above arguments we see that any functional completion of F must coincide with F and have the same norm and scalar product as F7.

It should be stressed that the second condition cannot be excluded from our theorem. We shall demonstrate this by the following example:

Consider the unit circle Iz I < 1. Take there an infinite sequence of points {, I~ such that

Z(-ZnI) < KO We shall denote by E the set of all points zn, and in E we shall consider

the class F of all polynomials in z. It is obvious that the values of a polynomial cannot vanish everywhere in the set E 'if the polynomial is not identically zero. Consequently the values of a polynomial on the set E determine completely the polynomial. We define the norm for a function f of the class F by the formula

lii2 ff p (Z) 12dxdy, z = x + iy,

where p denotes the polynomial whose values on the set B are giveni by the function f. We see that F satisfies all requirements for a Hilbert space with the exception of completeness. The first condition of our theorem is satisfied but the second is not. To prove the last assertion we take the Blaschke function 4(z) corresponding to {, z1. This function has the following properties

Jc(z)JI <1i fcr Iz I < 1.




The function 4(z) is a strong limit of polynomials Pk(z) in the sellse

rim f +(z) - pk(z) l2dxdy = 0.

Consequently the sequence { pk } is a Cauchy sequence in our class F and the polynomials Pk(z) converge at each point z,. to O(zS) = 0, in spite of the fact that the norms ||pkjj converge to ||f|l >o.

This example also shows us the significance of condition 20. We can say that if condition 20 is not satisfied it means that the incomplete class is defined in too small a set. If we added a sequence of points {I z' } of the unit circle with a limit point inside the circle to the set E of our example, then the class of polynomials, considered on this enlarged set E1 with the same norm as above, would satisfy condition 20. There are infinitely many ways of en- larging the set E where an incomplete class F is defined so as to insure the fulfillment of condition 20.

In the general case we can always proceed as follows. We can consider the abstract completion of the class F by adjunction of ideal elements. This completion leads to an abstract Hilbert space Q$. To every element of tI there corresponds a well determined function f(x) defined on the set (the limit of Cauchy sequences in F converging to this element). But to different elements of Xi there may correspond the same function f. The correspondence is linear and the functionalsf(y) are continuous in the whole '. To every point yGE there corresponds an element h, E, S such that f(y)- (, h,), where f is any element of 'D corresponding to the function f(x). As there are elements of &

different from the zero element and which correspond to the function identically zero on E, it is clear that the set of elements h, is not complete in the space SJ. (This is characteristic of the fact that condition 20 is not satisfied.) To the set of elements h, we can then add an additional set of elements so as to obtain a complete set in !5. This additional set will be denoted by E'. We can then extend the functions of our class F in the set E+E' by defining, for any element hlEE', f(h) (f, h). This class of functions, so extended in E+E', will then satisfy the second condition.

We shall complete this section by the following remark: It often happens that for the incomplete class F a kernel K(x, y) is kcnown such that for every y, K(x, y) as function of x belongs to F (or, even more generally, belongs to a Hilbert space containing F as a subspace), this kernel having the reproducing property

f(y) = (f(x), K(x, y)) for every f E F-

It is immediately seen that the first condition of our theorem follows from this reproducing property so that it suffices to verify the second condition in order to be able to apply our theorem.

5. The restriction of a reproducing kernel. Consider a linear class F of




functions defined in the set E, forming a Hilbert space and possessing a r.k. K(x, y). K is a positive matrix. If we restrict the points x, y to a subset E1CE, K will still be a positive matrix. Thlis means that K will correspond to a class F1 of functions defined in E1 with an adequate norm || fl1. We shall now determine this class F1 and the corresponding norm.

THEOREM. If K is the reproducing kernel of the class F of functions defined in the set E with the norm 11 11, then K restricted to a subset E1CE is the reproducing kernel of the class F1 of all restrictions of functions of F to the subset E1. For any such restriction, f' ? F1, the norm IlIf,il 1 i is the minimum of I If I for all f G F whose restriction to E1 is fi.

Proof. Consider the closed linear subspace FoCF formed by all functions which vanish at every point of E1. Take then the complementary subspace F'F=FFo. Both Fo and F' are closed linear subspaces of F and possess reproducing kernels Ko and K' such that

(1) K = Ko + K'.

Since Ko(x, y), for every fixed y, belongs to Fo, it is vanishing for xCE1. Consequently,

(2) K(x, y) K'(x, y), for x C E1.

Consider now the class F1 of all restrictions of F to the set E1. If two functions f and g from F have the same restriction fi in E1, f- g vanishes on E1 and so belongs to Fo. Conversely, if the difference belongs to Fo, f and g have the same restriction fi in E1. It is then clear that all the functions fE F which have the same restriction fi in E have a common projection fl' on F' and that the restriction off,' in E1 is equal also tofi. It is also clear that among all these functions, f, fl' is the one which has the smallest norm. Consequently by the definition of the theorem, we can write

(3) i = =

The correspondence between f' E F1 and f' e F' obviously establishes a one-to-one isometric correspondence between the space F1 with the norm || |1, and the space F' with the norm || ||-

In order to prove that for the class F1 with the norm 1 the reproducing kernel is given by K restricted to E1, we take any function fi C F1 and consider the corresponding function f ECF'. Then, for yEE1, fi(y) =1 (y) =(fo'(x), K'(x, y)).

Since K'(x, y), for every y belongs to F', we may now write f'(y) =(fl'(x), K'(x, y)) = (fi(x), K1(x, y))', where Ki(x, y) is the restriction of K'(x, y) (considered as function of x) to the set E1.

By hermitian symmetry we obtain from (2)




K(x, y) = K'(x, y), for every y ? El.

This shows that the restriction KI(x, y) of K'(x, y) coincides with the restriction of K to the set El, which completes our proof.

The norm|| li is especially simple when the subclass FoCF is reduced to the zero function. In this case F'= F, each function fiE F1 is a restriction of one and only one function fE3F (since every function of F vanishing in El vanishes identically everywhere in E), and therefore I|fi I = IIfI for the function f having the restriction fi.

6. Sum of reproducing kernels. Let Kl(x, y) and K2(x, y) be reproducing kernels corresponding to the classes F1 and F2 of functions defined in the same set E with the norms [ III and || 112 respectively. K, and K2 are positive matrices, and obviously K=K1+K2 is also a positive matrix.

We shall now find the class F and the norm || || corresponding to K. Consider the Hilbert space ii formed by all couples {fi, f2 } with fiCFt. The metric in this space will be given bv the equation

11 {fl, f2} Hf = -I- f212.

Consider the class Fo of functions f belonging at the same time to F1 and F2 (Fo may be reduced to the zero function). Denote by .So the set of all couples {f, -f} for fEzFo. It is clear that 1io is a linear subspace of S&. It is a closed subspace. In fact if {fni -fn } -- {f', f" } then fn converges strongly to f' in Fl, and -fn converges strongly to f" in F2. Consequently, fn converges in the ordinary sense to f' and -fn to f", which means that f" = -f' and f' and f" belong to Fo. St0 being a closed linear subspace of .S we can consider the complementary subspace &' so that 1- = &09GE'. To every element of S: {f', f" } there corresponds the function f(x) =f' (x) +f" (x). This is obviously a linear correspondence transforming the space S& into a linear class of functions F. The elements of & which are transformed by this correspondence into the zero function are clearly the elements of to. Consequently, this correspondence transforms &' in a one-to-one way into F. The inverse correspondence transforms every function fE F into an element { g' (f), g"(f) of S'. We define the metric in F by the equation

2lfl2 = I {g'(f), g"(f) }l12 = IIg'(f)I 1

+ IIg,(f)II2

Our assertion will be that to the class F with the above-defined norm there corresponds the reproducing kernel K = K1+ K2. To prove this assertion we remark:

(1) K(x, y) as a function of x, for y fixed, belongs to F. Namely, it corresponds to the element {Kl(x, y), K2(x, y) } CE&.

(2) Denote for fixed y, K'(x, y)=g'(K(x, y)) and K"(x, y)=g"(K(x, y)). For a function fE F, we writef' = g' (f), f" = g9 (f). Consequently, f(y) =f' (y)

+f"(y), K'(x, y) +K"(x, y) =K(x, y) =K,(x, y) +K2(x, y), and thus K"(x, y) -K2(X, y) = -[K'(x, y) -K,(x, y) ], so that the element { K1(x, y) -K'(x, y),




K2(x, y)-AK" (x, y) }IC So. Hence

f(y) = f'(y) + f"(y) = (f'(x), Ki(x, y))1 + (f"(x), K2(x, y))2

({f', f" }, { Kl(x, y), K2(x, y)})

-(({f',f }, {K'(x, y), K"(x, y)}) + ({f',f"}, {Kl(x, y) - K'(x, y), K2(x, y) - K"(x, y)}).

The last scalar product equals zero since the element {f', f" } CI ' and the element {Ki(x, y)-K'(x, y), K2(x, y)-K"(x, y) } ES(o. The first scalar product in the last member is, by our definition, equal to (f(x), K(x, y)) which proves the reproducing property of the kernel K(x, y).

We can characterize the class F as the class of all functions f(x) =fi(x) +f2(x) with fiEFi. In order to define the corresponding norm without passage through the auxiliary space, we consider for every fC F all possible decompositions f=fi+f2 with fiEFi. For each such decomposition we consider the sum I fj |2+IIf2I 2. IlfI| will then be defined by

|If 2 = min [1 + |12112] for all the decompositions of f. To prove the equivalence of this definition and the previous one we have only to remember that f(x) corresponds to the element {f,, f2} E&1 and also that f corresponds to {g'(f), g"(f)} E& ', that is, f =f1 +f2=g'(f)+g"(f). Consequently,

f2 - g"(f) = - [f1 - g'(f)]

so that {ff-g'(f), f2-g"(f) } ?(o and

1Ifil' I+f If2112 = {fl, f2} j2 = I{g'(f) g"(f)} 2 + {fl - g'(f), f2 -

and this expression will obviously get the minimal value if and only if fi=g'(f), f2=g"(f) and its value is then given by

I {gI(f), g"(f) }jj2,

which by our previous definition is IIf I2. Summing up, we may write the following theorem:

THEOREM. If Ki(x, y) is the reproducing kernel of the class Fs with the norm || |i, then K(x, y)=Kl(x, y)+K2(x, y) is the reproducing kernel of the class F of all functions f =fl +f2 with fi C Fi, and with the norm defined by

f|fl =min 1| + 1|f212],

the minimum taken for all the decompositions f =fj +f2 with fi Ez Fi ().

It is easy to see how this theorem can be extended to the case where

(6) This theorem was found by R. Godement [1 ] in the case of a positive definite function in a group.




K(x, y)= s Ki(x, y). A particularly simple case presents itself when the classes F, and F2 have no function besides zero in common. The norm in F is then given simply by 1fl 12 = Ij f112 +IJIf212.

In this case (and only in this case) F1 and F2 are complementary closed subspaces of F.

If we denote by F the class of all conjugate functions of functions of a class F, then the kernel of F7 is clearly Ki(x, y) =K(x, y) =K(y, x) (K is the kernel of F and in F the scalar product (f, k), is given by (g, f), the norm j|flll by l|ff1). Consequently Re K(x, y) =2-'(K(x, y)+K(y, x)) is the reproducing kernel of the class Fo of all sums f+g, for f and g in F with the norm given by

(1) ~ ~ ~~~~11=i 2 m [11f112 4- 1g112], (1) llfil~~o = 2 min ]J +lG ] the minimum taken for all decompositions 4 =f+g, f and g in F.

If F is a complex space corresponding to a real space, that is, if F= F and IlflI =llf1l (see ?1), it is clear that Fo= F and IfifII= |If1. Consequently the kernel K = Re K is real and this property characterizes the kernal K corresponding to a real space.

7. Difference of reproducing kernels. For two positive matrices, K1(x, y) and K(x, y), we shall write

1) K1 <K K

if K(x, y)-Kl(x, y) is also a p. matrix. From K<?K2?<K3, it follows clearly that K1<<K8. On the other hand, if

K1K<K2 and K2<<Kl, it follows that K1 = K2. In fact, we then have K2(x, x) - Ki(x, x) 2 0 and also Ki(x, x) - K2(x,x) 2 0, which means K2(x,x)- Ki(x, x) 0. Further, by the property of positive matrices, |K2(x, y)- Ki(x, y) 12 < [K2(x, x)- Kl(x, x)][K2(y, y)- Ki(y, y)] = 0,

so that K2(x, y) = Kl(x, y) for every x, y in E. Thus we see that the symbol << establishes a partial ordering in the class of all positive matrices.

THEOREM I. If K and K1 are the r.k.'s of the classes F and F1 with the norms | ||, | |l.and if K1<KK, then F1CF, and Ilfill_?lIfil ffor every fiCFj.

Proof. K1i<K means that K2(x, y) = K(x, y) -K (x, y) is a positive matrix. Consider the class of functions F2 and the norm |l 112 corresponding to K2. As K = K1+K2 we know by the theorem of ?6 that F is the class of all functions of the form f1(x) +f2(x) with f;E F, and f2E F2. In particular, when f2=0, the class F contains all functions fi C F, so that F,C F. On the other hand, in F we have, by the same theorem,

2 f'ff = mi [l|f,112 + |1f11 2] for all decompositions f' =fl'.+f2, with f) F, and f2' ECF2. In particular, for




the decomposition fi =fi +0, we obtain

lifll'2 <

which achieves the proof.

THEOREM II. If K is the r.k. of the class F with the norm , and if the linear class FlC Fforms a Hilbert space with the norm Hi Ill, such that f fi|

>-|fil Ifor every fi ? F1, then the class F1 possesses a reproducing kernel K1 satisfying KK1<<K.

Proof. The existence of the kernel for F involves the existence of constants M, such that jf(y) I _ JVIIfl | | for all f F. In particular, for f1 E F1 C F, wre get

I fi(y) ? < MS|fI1f I< M? | IJ 1

which proves the existence of the kernel K1 of F1. Let us now introduce an operator L in the space F, transforming the

space F into the space F1 and satisfying the equation

(fi, f) = (fi, Lf) 1, for every fi ? F1.

The existence and unicity of Lf is proved in the following way: (fi, f), for fixed f, is a linear continuous functional of fi in the space F. A fortiori, it is a similar functional in the space F1. As such, it is representable as a scalar product in F1 of fi with a uniquely determined element Lf of F1.

The operator L is everywhere defined in F, linear, symmetric, positive, and bounded with a bound not greater than 1. It is clear that L is everywhere defined and linear. It is symmetric because for any two functions f, f' of F, (Lf, f') = (Lf, Lf')1= (Lf', Lf)l= (Lf', f) = (f, Lf'). It is positive because (Lf, f) = (Lf, Lf), _ 0. It is bounded with a bound not greater than 1 because (Lf, f)=(Lf, Lf)1=I LfJ2>lLffl2. Consequently, |Lffl2<(Lf, f) ?||Lf||. ||f|, and thus ||Lf | < 1ffl.l

Consider now the operator I-L (I being the identical operator). This operator clearly possesses the same properties as those enumerated above for L. Therefore there exists a symmetric, bounded square root L' of this operator. (In general there will be infinitely many L' available and we choose any one of them.) Hence

L-2= I - L.

We define F2 as the class of all functions f2=L'f for fCF. Denote by Fo the closed linear subspace of F transformed by L' into 0, and by F' the complementary space F9Fo. The functions of Fo are also characterized by the fact that L'2f = O (from L'2f = O it follows that (L'2f, f) = (L'f, L'f) =ffL'f 2 =0) which is equivalent to f = Lf.

Now denote by P' the projection on F'. Every two functions f, g of F transformed by L' into the same functionf2 of F2 differ by a function belong-




ing to Fo. Consequently, they have the same projection on F'. It is also clear that the class F' is transformed by L' on F2 in a one-to-one way. We remark further that

F2 C F'

since for foC Fo, (fo, L'f) = (L'fo, f) = (0, f) = 0. In the class F2 we introduce the norm 2 by the equation

|Jf2f|2 = |IP'ffl. for any f with f2 = L'f.

With this metric the class F2 is isometric to the subspace F'CF (the isometry being given by the transformation L'). It follows that F2 is a complete Hilbert space and we shall show that the class F2 with the norm || 112 admits as reproducing kernel the difference K(x, y) -Kl(x, y). To this effect we remark firstly that the operator L is given by the formula

fi(y) = Lf = (f, Ki(x, y)).

In fact, since Lf Fl, fi(y) = (fi(x), Kl(x, y))i= (f(x), Kl(x, y)). Conse- quently for any fixed z, the operator L applied to K(x, z) gives the function LK(x, z) =Ki(y, z). If follows that the operator I-L =L'2 transforms K(x, z) into L'2K(x, z) =K(y, z) -Ki(y, z). This formula proves, firstly, that K(x, z) -Kl(x, z) as a function of x belongs to F2, since it is the transform of L'K(x, z) by L'. Secondly, we prove the reproducing property for any f2?F2:

12(y) = (f2(x), K(x, y)) = (L'f, K(x, y)) - (f, L'K(x, y)) = (P'f, P'L'K(x, y)) = (L'f, L'L'K(x, y))2

= (f2(x), K(x, y) - Ki(x, y))2.

In these transformations we took f as any function of F such that f2 = L'f and we used the property L'K(x, y) E F2C F'. This finishes the proof of our theorem.

The proof of Theorem II has established even more than the theorem an- nounced: namely, it gives us the construction of the class F2 and the metric

ff 112 for which the difference K-K1 is the reproducing kernel. Let us sum- marize this in a separate theorem.

THEOREM III. Under the hypotheses of Theorem II, the class F2 and the norm || 112 corresponding to the kernel K2= K - K1 are defined as follows: the equation fi(y) Lf = (f(x), Ki(x, y))z defines in F a positive operator with bound not greater than 1, transforming F into F1 CF. We take any symmetric square root L'= (I-L)1/2. F2 is the class of all transforms L'f for f F. Let Fo be the closed linear subspace of all functions f F with f= Lf and let F' be Fe Fo. L' establishes a one-to-one correspondence between F' and F2. The norm IIf2 2 for f2=L'f', f'CF', is then given by




IIf2II2 = VlTl.

The construction of the class F2 is somehow complicated because we need the square root L' of the operator I-L. However, there exists a much easier way of constructing an everywhere dense linear subclass of F2 and the norm || 112 in this subclass. Namely, we can consider the class F2' of all transforms (I-L)f =f-Lf = L'2f, of f E F, by the operator I-L.

This clearly is a linear subclass of F2. The norm || 112 in Fe can be defined as follows:

IJf'|2 = IL'f2 (L'f, L'f) = (f, L f) = (f, f - Lf) = (f, f) - (f, Lf) = (f, f)- (Lf, Lf)1.

In these transformations we considered f2' = f-Lf = L'f. F2' is everywhere dense in F2 (in respect to the norm || 112), otherwise we

would have a functionf2=L'f'z0 with (L'f', L'2g)2=0 for all gGF. We can suppose here f'eF' so that 0= (L'f', L'2g)2=(f', L'g) = (L'f', g) whence L'f' 0, f' C Fo, which is impossible.

The simplest case is the one where F2 = F2. By using the spectral decomposition, we can easily show that this case presents itself if, and only if, the zero is not a limit point of the spectrum of L', which is the equivalent of saying that 1 is not a limit point of the spectrum of L. In this case F2 = F'. If L has a bound <1, the spectrum does not contain 1 and the subspace F' coincides with the whole space F so that F2 = F.

When L is completely continuous, the only limit point of the spectrum of L is zero, so that I is certainly not a limit point, and F2' = F2 = F'.

We shall add still another theorem which results immediately from Theorem 1Il.

THEOREM IV. Let K be the r.k. of class F. To every decomposition K=K, +K2 in two p. matrices K1 and K2, there corresponds a decomposition of the identity operator I in Fin two positive operators L1 and L2, I= L1 +L2, given by

Lif(y) = (f(x), Kl(x, y)), L2f(y) = (f(x), K2(x, y)),

such that if Lu/2 and L112 denote any symmetric square roots of L1 and L2, the classes F1 and F2 of all transforms L12f and L'/2f respectively, f C F, correspond to the kernels K1 and K2. If Fio, i = 1, 2, is the class of all f EF with Lif = 0, and if Fi"= FFio, then L/2 establishes a one-to-one correspondence between FR" and Fi and the norm i| ||? in Fi is given by I1L /2f lI i X=-fI1 for every f C Fi' .

Conversely, to each decomposition I=L1+L2 in two positive operators there correspond classes Fi with norms 1I 1I i defined as above. The corresponding r.k.'s Ki are defined by Ki(x, y)=LiK(x, y) and satisfy the equation K=K1+K2.

8. Product of reproducing kernels. Consider two positive matrices K1




and K2 defined in a set E. Using a classical result of I. Schur concerning finite matrices, it is easy to prove that their product K1j K2 is also a positive matrix. We shall prove this theorem, constructing at the same time the class F and the norm | |corresponding to the matrix K=KK1* K2(7). To this effect, we consider the class Fi and the norm f| fi corresponding to Ki and form the direct product F' of the Hilbert spaces F1 and F2, F"= F10F2(8). We construct this direct product in the following manner: We form the product set E'=EXE of all couples of points {x1, x2}, xiCE. In the set E' consider the class of all functionsf'(xi, X2) representable in the form:

n (kc) (L.)

(1) f'(xl, X2) = E.fl (xl)f2 (X2), k=1

with fk) C F1 and fA) F2. As scalar product of two such functions we define: n m (k () (A;) (I)

(2) (f1' g')' = E (fl g )1(f2 (Ek 2 )2, k=1 1=1

where mn is the number of terms in the representation of g'. The same function f' may admit of many different representations of type (1). The scalar product, (f', g')', is independent of the particular representation chosen for f' and g'. In fact, we see immediately from (2) that

in (1) (1) (3) (f, g')' = ((f(xl, X2), gl (X1)) , g2 (X))2

1=1

which proves that (f', g') is independent of the particular representation off'. In a similar way we prove that it is independent of the particular representation of g'. We still have to prove that (f', g')' satisfies all the requirements for the scalar product. It is clearly seen that it is a bilinear hermitian form in f', g' and it remains to be proved that (f', f')' ?0 and is equal to zero only when f'=O. In order to prove this, we take any representation of f' of type (1) and orthonormalize the sequences {f()} and {f2t)} in the spaces F1 and F2 respectively. Denote by {ft`l) } and {f(lul) } the orthonormalized sequences, where k= 1, 2, . . . I n, I= 1, 2, . . *, n2.

Every function f,k) is then a linear combination of the orthonormal functions f(k'l) so that we obtain a representation for f' as a double series

ni n2 (,1 ,1 (4) f'(Xl, X2) = E E ak,lfl f2

k=1 1=1

We then obtain for (f', f')' the following expression

(7) The idea of the proof was arrived at independently by R. Godement and the author. Godement applied it only to positive definite functions.

(8) For the notion of direct product of abstract Hilbert spaces see J. v. Neumann and F. J. Murray 1f].




(0 n 02 )0 70 2 (kh i) (h',1) (1, 1) (1' 1) (f',f) =I12X ak,z&h ',z(f ,fi )1 2 f~ )2

(5) k=l I-1 k'-l V-I n 1 0n2

~~~~~2 = t L2Iak,lI2 k=1 1=1

It is clear from this representation that (f', f') > 0 and equals zero only when all the Cak,1=0, that is, whenf'=O. The class of all functionsf' of type (1) does not yet form in general a Hilbert space because it may not be complete. To complete this class with respect to the norm || jj', we consider a complete orthonormal sequence {ghk) } in the space Fi, i= 1, 2. It is obvious that the double sequence { g )(xi) g2) (X2) } is composed of functions of type (1) and is orthonormal in respect to the norm |1' Consider then, all the functions g' of the form

00 00

(6) g'(x x2) - , E a;,zgi (X1)g2 (X2) k=I 1=I

with 00 00

(7) Q', g')' I akE 1 12 <

It is clear that any finite sum of type (6) is also of type (1) and that the norm || 11' for such finite sums coincides with the norm introduced in (7). We prove firstly that every sum of type (6) is absolutely convergent for every xi, X2. In fact, as the class F1 possesses the reproducing kernel K1, in view of (7) we have

ak, g1 X1) |< [KIlX1, X1) ]112[ E ak, 1 12 k=1 k=1

Then, 00 t00k

S: l !,zJ g' (xi) gI (X2)

0 00 1/2

(8) g2 (X2) [K1(x1, X1) ]112 [ 2 ak, 1 I=1 h=1

- oo oo -1/2

-< [K (Xl, xI)] ]1/2 [K2(X2t X2) ]1/2. E Ea|k,l 12 k=1 1-1

since the space F2 possesses the reproducing kernel K2. The class of functions g' of the form (6) clearly forms a complete Hilbert

space, isomorphic with the space of double sequences {chk, } satisfying (7). The inequality (8) gives us further, for a function g' of type (6),




I g'(y, y2) 1 < K]I(y1, yl)'I2IK2(y2, y2)1/2 lgj1,

which means that the space of functions of type (6) possesses a reproducing kernel.

It remains to be proved that this space is the completion of the class of all functions f' of type (1) with the iiorm || |'. As already the finite sums of type (6) are everywhere dense in the space of all functions of type (6) it is sufficient to prove that every function of type (1) is of type (6). For this, it is enough to prove that every function of type (1) may be approximated as closely as we wish (in respect to the norm fl II') by finite sums of type (6). Let us consider a representation (1) of the function f'. We can approximate every f.) by a finite linear combination h?) of functions gi() so that 11hOt)I < Iffl i, |flf)- h(k)fl i < e. Before we proceed further, we shall prove for everv function f' and any of its representations (1) the inequality

< E |lf 1)l |1 - |jf2("j 12- k=1

In fact,

(1) (k) (1)

11fI,1 =(f f) = Z E (jf ) fl)' ()1 U f2 )2 7.=1 1=1

<: E || 1 ||1 ||f ||1 * | | 12 | | 2 fE2 1|1|2 k=1 1=1

= [ z|fl || |1f2 1|2 k=l

Continuing with the proof of the approximation we consider the functions

h' ( ) E h ( k) ( k ) lz(X 1, X2) = hik)(Xi)f~k (X2), 1=1

g'(xl, x2) = E h (xl) h2 (X2) k=1

It is clear that h' is of type (1) and that g' is at the same time of type (1) and (6), which can be seen by developing the functions h(k) as linear combinations of gAl). Denoting by M the maximum of all I1f(|Ii, we obtain

|lf' - g'II' < I I' - h'I' + 1|'I - gl ',

Ift II/ll = E I11 (f '

(Xl) - II` (Xi))f2 (X2)1 I'

k=1

n -k Iz(k)I II(k)I AL= < E lc=1 hi )||l-|| f 112 Me nME,

k=31 k=l




1h' - g|'= E (x1)((fI (x2) - h2 (X2))

>< E 11 h (k||1)||f - h2 ?12 -E Ale = nM-E. k=l k=l

Finally, we obtain

- g'1' < 2nME,

whichi proves our assertion. The class of all functions of type (6) with the norm given by (7) forms

the direct product F' = F10 F2. As it is obtained by functional completion of the class of functions of type (1), this class being independent of the choice of orthogonal systems { gI } and { g() }, the class F' is also independent of the choice of these systems (see the uniqueness of functional completion in ?4).

We shall now prove the following theorem:

THEOREM I. The direct product F' = F1 0 F2 possesses the reproducing kernel K'(xi, X2, Yl, y2)= K1(x1, y1)K2(x2, y2).

The proof is immediate. Firstly, as a function of xi, x2, K' is of the form (1) and so belongs to F'. Secondly, for any function g' of the form (6) we have

(k) (1) g'(yl, y2) = > E ak,I(gl (xi), Ki(xi, yl))1(g2 (X2), K2(x2, y2))2

= (g'(XI, X2), K'(x1, X2, yl, y2))'

which completes the proof. From Theorem I we see immediately that the kernel K(x, y)

=Ki(x, y)K2(x, y) is a p. matrix as the restriction of the kernel K'(xi, x2, yl, y2)

to the subset E1CE' consisting of the "diagonal" elements {x, x} of E'. Further, from the theorem of ?5 we obtain the class of functions and the norm corresponding to the kernel K. Thus we have the following theorem.

THEOR:EM II. The kernel K(x, y) = K, (x, y)K2(x, y) is the reproducing kernel of the class F of the restrictions of all functions of the direct product F' = F1 0 F2 to the diagonal set E1 formed by all the elements { x, x} ICE'. For any such restriction f, I|f|| = min |g'| 'for all g' F', the restriction of which to the diagonal set E1 is f.

RETMARK. Let { g() } be a complete orthonormal system in F,. Then every function fE F is representable as a series

f(x) f2 (x)g i(x), f2) E F2, 2 < (k

1 1

Among all such representations of f(x) there exists one (and only one)




which gives its minimum to the sum EIIf2 II2. This minimum is equal to Lffl 2. We can apply Theorem II to a class F and its conjugate F. The product

of the corresponding kernels is K(x, y) 2=K(x, y)K(y, x) and the corresponding class may be obtained from the remark above.

9. Limits of reproducing kernels. We shall consider two cases: (A) essentially, the case of a decreasing sequence of classes F1DF2D with a decreasing sequence of kernels K1>>K2>>K3>> ; (B) essentially, the case of an increasing sequence of classes and kernels.

A. The case of a decreasing sequence. Let { En } be an increasing sequence of sets, E their sum

(1) E-E1 + E2 + ***,E1 C E2 C**

Let Fn, n= 1, 2, - , be a class of functions defined in En. For a function C F,, we shall denote by fnm < n, the restriction of fn to the set Em CEn

(fnn =fn). We shall suppose then that the classes F, form a decreasing sequence in the sense

(2) for every f, C Fn and every m < n, fnm C Fm.

Suppose further that the norms || IIn defined in Fn, form an increasing sequence in the sense

(3) for every fn CEFn and every mn< n, I|fnmllm - Ifn|In.

Finally, we suppose that every Fn, possesses a reproducing kernel Kn(X, y). The case of all sets En equal, E1 = E2= - * = E, is not excluded. Clearly,

in this case fnm =fn, Fn C Fm, and, following Theorem II of ?7, it is enough to suppose the existence of Kl(x, y) in order to deduce the existence of all IKn and to obtain the property Kn<<Km for m <n.

In the general case we have to introduce the restrictions Knm of Kn to the set Em (m < n). By the theorem of ?5, Knm is the r.k. of the class Fnm of all restrictions fnm for fn ? Fn. The norm in Fnm is given by

llfnrnHnm = min IIfn|Kn for allfi withfnm = fnm.

From (3) we get

llfnm||lnm >_ ||fn?n||JIm

and consequently, by Theorem II of ?7,

(4) Knm << Km, m < n.

We shall now prove the following theorem.

THEOREM I. Under the above assumptions on the classes F7, the kernels Kn converge to a kernel Ko(x, y) defined for all x, y in E. K0 is the r.k. of the class Fo of all functions fo defined in E such that 1? their restrictions fon in En




belong to F, n = 1, 2, * * *, 20 lim7,= I|fon|In <oo . The norm of fo CFo is given by jjfOj|O=1ir1n=o |lfon|In.

REMARK. Condition 10 implies, following (3), that limn=oIIfonI|n exists, but it may be infinite.

Proof. The convergence of Kn to K is to be understood in this way: any two points x, y in E belong to all En starting from some Eno on. Consequently Kn(x, y) are defined for n>no and we have to prove limn=w Kn(x, y) =K(x, y).

For fixed yCEk and k <m <n we have, following our assumptions,

IIKmk(X, y) - Knk(X, y)|k=< |Km(X, y) - Knm(X m, )n

= Km(y, y) - Knm(y, y) - Knm(y, y) + IlKnm(X, Y)jjm

? Km(y, y) - 2Kn(y y) + J|Kn(x, Y)11

= Km(y, y) - Kn(y, y).

From (4) it follows that Km-Knm is a p.d. matrix. Therefore Km(y, y) -Knm(y, y)=Km(y, y)-Kn(y, y)_O, and the sequence {Km(y, Y)}m.2 k is a decreasing sequence of non-negative numbers. Consequently it is a convergent sequence. (5) shows then that the functions Kmk(X, y) EFk, for fixed k and m-*cc, converge strongly in Fk to some function ?Ik(X) Fk. This involves limm=. Kmk(X, y) =limm=. Km(x, y) = $k(X) for every xCEk.

Since for every x, y in E we can choose a k so that x and y belong to Ek, it is clear that Km(x, y) converge and that the limit Ko(x, y) does not depend upon the choice of k. The function Ok(X) is clearly the restriction Kok(X, y) of Ko to Ek. Consequently Kmk(X, y), for fixed y, converges strongly in Fk to Kok(x, y) which belongs to Fk. We then obtain from (5), by taking n-> oo,

(6) flKmik(X, y) - Kok(X, y)||k < Kn(y, y) - Ko(y, y); flKok(X, y)|Ik ? |KOk(X, y) - Kmk(X, y)|k + H|Kmk(X, Y)|jk

< (Km(y, y) - Ko(y, y))1/2 + ||Km(X, y)lm

< (Km(y, y) - Ko(y, y))1/2 + (Km(y, y))1/2

and for m->oo, Kok(x, y)k =< Ko(y, y). Therefore for each yCE, Ko(x, y), as function of x, belongs to the class Fo

of our theorem. Let us now prove that the class Fo is a Hilbert space. Fo is linear, since I I afOn +I30nj I n-< I a I I IfOnj I n + I | | | gOn I n

0foI is a quadratic form, since

||4oO + lgol|o = lim |lafOn + [gonH|n n= ro

= lim [aall|fOn||n + a58(fOnt gOn)n + &:(gOn, fon)n + 05,BIgOn| n] n= oo




Since the quadratic form in square brackets converges for all values of the complex variables a, A, it converges to a form of the same kind. This proves that IIfoIJ2 is a quadratic form and also that

(7) (fo, go)o = lim (fo., gon)n. ?I= 00

It remains to be proved that Fo is complete. Take a Cauchy sequence {f() } CFo. The inequality Ifok) _f0k)H

(<infoi)-fo1)||I, for lI>k, gives, when l- oo, lf(om;-f(nH)11,<l(om)-f(n)ll Hence, {f0jk)I m=1,2, .. is a Cauchy sequence in Fk, limm.=O f(Om) ='kCFk. It is clear that 1lk iS the restriction to Ek of a function 'po defined in E. We have

(in 1. j(m -f ? II fr(n) I I I fok - Ok = ln hOk Ok 0 -

n=xo n=x0

This allows us to prove that tOC Fo, since it gives a bound for k|v/Okflk independent of k, namely

I|POkfl k ?< |jfok ||k + lim j|m) - fo ll o -< o I o + lrn j|iO - jf ll0o n=x0 n=xo

On the other hand, it shows that

I If (M) - 4,011 o = lim Ijf" - l I k _< lim fm - fo llo

k= oo n= o

and consequently limm=~o |f|(m)-/oIIlo=o. This achieves the proof of completeness.

We have still to prove the reproducing property of Ko. To this effect take anyfoEFo, yEE. For sufficiently large n we have

fO(y) = fOn(y) = (fOn(X), Kn(X, y))n = (fOn(X), KOn(X, y))n

+ (fOn(X), Kn(X, y) - K0n(X, y))n.

For n-> oo, the first scalar product in the last member converges, by formula (7), to (fog Ko(x, y))o. The second scalar product converges to 0; in fact, by formula (6) (with k = m = n), it is in absolute value smaller than

||fon|hnK,(x, y) - KOn(x, Y)I|n_< IfoIIo(Kn(y, y) - Ko(y, y))l/2

This achieves the proof of our theorem. B. The case of an increasing sequence. Let {En} be a decreasing se-

quence of sets, E their intersection

(8) E = E1*E2* ** a , E1 D E2 D*

Let Fn be a class of functions defined in En. As before, we define the restriction fnm, for fnEFn, but now m has to be greater than n. We suppose then that Fn form an increasing sequence




(9) for every fn C Fn and every m > n, fnm C Fm.

We suppose further that the norms j| |1, form a decreasing sequence

(10) for every fn C Fn and every m > i t, fnm|m ^ Lfnlnl

Finally, we suppose that every Fn possesses a r.k. Kn(x, y). Now, even for all En equal, we cannot deduce the existence of all kernels

Kn from the existence of one of them. As in the case A, we get for the restrictions Knm of Kn the formula

(11) Knm << Km form > n.

For yCE, { Kn(y, y) } is an increasing sequence of positive numbers. Its limit may be infinite. We define, consequently,

(12) Eo = set of y E E, such that Ko(y, y) = lim Km(y, y) < cc. M=W

For an illustration of this point consider Bergman's kernels Kn for a decreasing sequence of domains En. If the intersection E of the domains En is composed of a closed circle with an exterior segment attached to it, the set Eo will be composed of all interior points of the circle.

We suppose that Eo is not empty and define the limit-class of the classes Fn in the following way: let Fo be the class of all restrictionsfno of functions

fnCFn (n=1, 2, . . . ) to the set Eo. From (10), we know that the sequence { I fnk| k } k==n, n+?1, .. is decreasing and we can define

(13) l|fnO|lo = lim |fJnkjIk- k = xo

As in case A we prove that Aiflo'I2 is a positive quadratic form. This form is positive definite since from iifnoIIo = 0 it follows that for any yEEo, Ifno(y) I = Ifnk(Y)I = (fnk(x), Kk(x, y))k| -I<IfnkIjk(Kk(y y))1/2 f>|fnoIIO(Ko(y, y))l/2=O0 that is, fno =0. Consequently II Ilo is a norm in Fo.

In general Fo will not be complete. In order that Fo admit of a functional completion with a reproducing kernel, there are two conditions to be fulfilled which are given in the theorem of ?4. The first one is that for every y there exists a constant M, so that

(14) | fno(y) ? My Ifn0l 0 for all fno E Fo.

Let us remark that the functions fno may be considered as defined in the whole set E (taking the restriction offn to E). Then, for every yzE, the condition (14) is equivalent to Ko(y, y) = lim Kn(y9 y) < oo . In fact, from the latter condition it follows in the same manner as above that Ifno(y) I j< |f0oll o(Ko(y, y)) 1/2, that is, (14) with My= (Ko(y, y)) 1/2. From (14), by taking f.(x)=Kn(x, y) we get Kn(y, y)<MylIKno(x, y)l|o<Myl Kn(x, Y)iIn = My(Kn (y, y)) 1/2, that is, Kn(y, y) < M2.



366 N. ARONSZAJN (May

Consequently, condition (14), that is, condition 1? of ?4, is assured by our restriction to the set Eo. But the condition 20 of ?4 is in general not assured.

We may illustrate this by the counter-example of ?4. Take all the E. equal to the set E of this counter-example. As class Fn we take the n-dimensional subspace of the class F introduced there, consisting of all polynomials of degree <n. Each Fn is a complete space and its r.k. Kn converges to the Bergman kernel of the circle, restricted to E. Consequently Eo = E, Fo = F and || o is clearly the norm introduced there.

To overcome this difficulty we can proceed as indicated in ?4. We complete Fo by ideal elements; in the completed Hilbert space Fo we choose an additional set E' of ideal elements such that the functions of Fo, extended to Eo+E', with the same norm as in Fo, form a space admitting a functional completion leading to a class Po with a reproducing kernel Ko.

Following the theorem of ?5, we can return now to our set Eo by restrict- ing the functions of Po to Eo. If we take in the restricted class the norm defined in the theorem of ?5, we shall get as r.k. the restriction of Ko to Eo. The restricted class Fo* and its norm 11 llo can then be described, in terms of the space Fo and its norm | |lo, in the following way.

fo Fo* if there is a Cauchy sequence {ff() } CFo such that

(15) fo (x) = lirn Jo )(x) for every x E Eo,

(n) (16) 0 f (x)H= m rn lim lfo 110,

n=x

the minimum being taken for all Cauchy sequences {ffn) } CFo satisfying (15). There exists at least one Cauchy sequence for which the minimum is attained. Such sequences will be called determining fo*.

The scalar product corresponding to || Ilo is defined by

* ** ~~(n) (n) (17) (fo, go)o = lrm (fo , go ) o

n= xo

for any two Cauchy sequences {f8) } and { g()} determining fo* and go'. It is important to note that formula (17) is still valid when only one of

the sequences {f(on) }, { g(n)} is determining, the other satisfying only (15). All these facts about the space Fo* and its norm and scalar product are

easily obtained when we form, as in ?5, the subspace PoCPO of all f0o o vanishing in Eo. The complementary subspace PFo = Po-Fo is then in an isomorphic correspondence with Fo* f6' fo*, where f * is the restriction of f0' to Eo. Further,

0I = Vf10, fo 9O)o = (fo' 9o0)0




where jj |lo and ( , are the norm and scalar product in Po. A Cauchy sequence {f: } C Fo converges in Fo to a function fo. In the set

Eo it converges everywhere to a function f E Fo* which is the restriction of fo to Eo. fo* is also the restriction of Jo = projection of fo on P0'. Since fo-fo' E F-o we get

lir IIfn112 = 1lfoll 2 = -IIJ.o,I2 + lIf o - I'lL2 > If0*Il*o n?Woo

The equality here is attained if {fSn) } converges in Fo tofo. Such a Cauchy sequence we have called as determining fo,. If now two Cauchy sequences {ffo) } and { g(n } converge everywhere in Eo to foj and g *, they converge in Fo to vectors fo and go whose restrictions to Eo are fo* and g *. If jo and gJ are projections of fo and go on F', then (fo*, g0*)0* =(fo, g' )O, but lim I g()) - (fo, g0)0 . If one of the sequences, say {f0n) } }determines its limit in Eo, then Jo =fo, and (fo*, gO o, g )O = (fe'I go)O = lim(f( ), gOfl)) o

The space Fo* being completely defined we prove the following theorem.

THEOREM II. The restrictions Kno(X, y) for every fixed y EEo form a Cauchy sequence in Fo. They converge to afunction K *(x, y) C Fo* which is the reproducing kernel of Fo.

Proof. By an argument similar to the one used in (5) we obtain for

(18) I|Kmk(X, y) - Knk(x, y)|| < Km(y, y) - K.(y, y).

Taking k-> co, we have

(19) f|Kmo(x, y) - Kno(x, Y)||? < Km(y, y) - Kn(y Y).

This proves, together with (12), that {Kno(X, y) } is a Cauchy sequence in Fo. By property (14) this sequence converges for every xEEo to a function Ko(x, y) which, by definition (15), belongs to Fo.

It remains to prove the reproducing property of K:. To this effect take any fo E Fo and a Cauchy sequence {Jfo) } C Fo determining fo*. Each fo() is a restriction of somefk.CFk., fAn) =fkno. By (13) there exists an increasing sequence MI1<M2 < * such that

(20) mn > kn, Ilficnmnlml - Ifk_o01 - 1 n2

It is clear that {Kmno(X, y) } is also a Cauchy sequence converging to Ko(x, y). Consequently, from (17) it follows that

(21) (fo(x), Ko(x, y))0 = lim (fkno(x), KmnO(X, Y))o. n= oo

We may now write




(fk,o(x), Kmno(X, y))o = (fknmn(X), Kmn(X, y))mn

(22) - [(fk.m.(X), Km.(x, y))m. - (fkno(x), Km.o(x, y))o].

The square bracket is of the form [(g, h)mn -(go, ho)o] for g, h of Fmn (go, ho are restrictions of g, h to Eo). This is a bilinear form in g, h and the corresponding quadratic form (g, g)nm- (go, go)o =IIgll- _IIgo|1| is positive (following (10) and (13)). Consequently the Cauchy-Schwarz inequality is valid for this form and in the case of the square bracket of (22) it gives in connection with (20)

t [ * * * ] |< [Ifknmn In - IlfknolO1 [ nf Kmn(X, y)1|n - IfKmno(X, Y)A1]

1 1 - Kmn(X, y)lmn = Kmn(y, y)'/2. n ns

For n-* oo this converges to 0, since Kmn(y, y) IKo(y, y) < oo. Therefore, (21) and (22) yield

(f*(x), Ko(x, y))0 = lim (f nmn(X) IKmn(X,

Y))mn = lim fknmJ(y) n=xo n=xo

= limfknO(y) = lim fj (y) = fo(y),

which is the reproducing property of KO. REMARK. A particularly simple case is the one where the class F0 with

the norm I| 11o happens to be a subspace of a class F possessing a reproducing kernel. Then, condition 2? of ?4 is clearly satisfied; FO is the functional completion of Fo, the norm || Ilo is an extension of the norm 11 11, and FO' is simply the closure in F of Fo.

A trivial case of this kind is one where the Fn form an increasing sequence of subspaces of a class F with a r.k. Hence E1= E2= = Eo= and F* is the closure of the sum EFn.

10. Construction of a r.k. by resolution of identity. Let us give a brief resume of the essential properties of resolutions of identity in a Hilbert space . (for a complete study of resolutions of identity, especially ifn connection with the theory of operators, see M. H. Stone [1]). For simplicity's sake we shall suppose here that the space .S is separable.

We call a resolution of identity a class { Px } of projections in .&, depending on a real parameter X, - oo <X < + co, and having the following properties:

1. P, is a projection on a closed subspace t>.C&, increasing with X: !&x'C.)x for X<X.

2. Px-*0 for X-o- ; Px->I (identity operator) for X->+ oo. For any open interval A: X'<A<X", we define

(1) A,S = tx', AP = PX,,- Px,.




AP is the projection on MA'. For any decomposition of the real axis into intervals Ak = (Xk, xk+1),

- * * * <X-2 <X-1 <XO <-X1 <X2 < * * + ??, we have, obviously, (2) I=ZAkP,

k

the series converging in the sense of strong limit for operators. A real number X0 belongs to the spectrum of { Px } if APO for every

interval A containing X0. The numbers belonging to the spectrum form a closed set.

For any real 0 and any decreasing sequence of intervals An containing 0 and converging to 0 there exist the limits

(3) bP = lim A"P, b0og = lim An&)s

the second limit being the intersection of the decreasing sequence of subspaces a These limits do not depend on the choice of An and 30P is the projection on b Only for an enumerable set of 0, say {Gk }, is 30P$?O.

If we have 5o,,P=I, which means 601&4-82+-** =g, we say that the spectrum of {Px} is discrete.

If, for all 0, 3oP= 0, we say that the spectrum of { P } is continuous (often called purely continuous).

In our applications we shall meet, essentially, only discrete spectra or continuous spectra.

It has been proved (theorem of Hellinger-Hahn) that for any spectrum there exist finite or infinite systems {f,,(X) } of elements fn(X) C, depending on X, such that if we denote by Afn the difference f (X") -fn(X'), we have

(a) for m n, (Alfm, A2fn) = 0 for any intervals A,, A2. (b) (A if., A2fn) =0 for any non-overlapping intervals A1, A2. (c) For every interval A, the elements Alfn, n = 1, 2, * * * for all A1 CA, belong

to the subspace A\& and form a complete system in MA'. The minimal number of elements in such a system {f,)(X) } is called the

multiplicity of the spectrum. The spectrum is called simple if the multiplicity 1, that is, if there exists such a system with only one element fl(X). In the case of a discrete spectrum the multiplicity is the maximal dimen-

sion of the subspaces b If a system of elements {f,(X) } satisfies (a) and (b) and instead of (c)

satisfies the weaker condition (c') The elements Afn for n = 1, 2, * * * and for all intervals A form a complete

system in 6&, then the system {fn(X) } determines a corresponding resolution of identity Px } (for which it satisfies (c)) in the following manner:

Sxi is the subspace generated by all the Af, n = 1, 2, * * , A = (', MX ") with x,,<x.




This resolution of identity is continuous to the left, that is, - = lim &;, for X'/X. This condition (or the right side continuity) is usually accepted, for reasons of convenience, as an additional condition on resolutions of identity.

For any system {f,,(X) } satisfying (a), (b), and (c), it is seen that I |fn (X) | 2 is a non-decreasing function .n(X), A/\zn=/Un(X") -Un(X') =11/fnJ12. We consider the measure gn, introduced on the real axis by .n('X), which leads to the Lebesgue-Stieltjes integral fJ?(X)d/un(X). It has been proved that for every uCSi there exists the limit

(U, Afn) _ d(u, fn(X)) (4) 4/)n(XA) = X,olim,,\X -XAf'112 dAn(X)

for all X with exception of a set of gUn-measure 0. We have further

A00 ,5 1JU112 I ....O /n(X) 12d/.n-

n _oo

(6) (u, v) = E fn((X)/n(X)dJUn, where n(X) = ( f()) n -oo d/.dn(X)

Let us now apply the above considerations to the construction of a r.k. We suppose that our Hilbert space & is a class of functions defined in E with a r.k. K(x, y).

For a given resolution of identity { P, } every subspace A,\ will have a r.k. which we shall denote by AK(x, y). The kernel AK determines the projection AP by the equation

(7) APf = fi(y) = (f(x), AK(x, y)), for any f E '.

The kernel K corresponds to the identity I and following (2) we have

(8) K(x, y) = EAkK(x, y), k

for any decomposition {Ak} of the real axis. The series in (8) converges absolutely. In fact, following (2), the series K(y, z) = IK(y, Z) = ZAkPK(y, z) = Z(K(x, Z), AkK(x, y))x= ZAkK(y, z) as function of y is strongly convergent. It converges then in the ordinary sense for every y, in particular for y=z. Thus, K(z, z) = ZAkK(z, Z) < 0, AkK(z, z) _0. Consequently

Z AkK(x, y) | < E (AkK(x, X))112(AkK(y, y))112

< [ZAK (x, x) ZAkK(y, y)]12

If the resolution of identity P, } has a discrete spectrum {Ok} and if the r.k. of b is denoted by bokK, then we have again an absolutely convergent representation




(9) K(x, y) = E 60oK(x, y).

An especially important case which is most often applied is one where the spectrum is simple. Then the subspaces bok, are one-dimensional and each is generated by a function gk(x) which we can suppose normalized, ||gk||=1. The functions gk(x) form a complete orthonormal system in t. The kernels 3okK(X, y) are given by gk(X)gk(y), and (9) takes the form of the well known development of the kernel in an orthonormal complete system

(10) K(x, y) = E gk(X)gk(y), k

which for a long time was taken as a basis of the definition of a r.k. Suppose now that the resolution of identity { P } is given by a system of

functions fn(X) -fn(x, N) satisfying (a), (b), and (c'). Following (4) we define for u = K(x, y) (considered as function of x), the functions

(K(x, y), Afn) 1 fn(y, X") - fn(y, X') dfn(y, X) (11) In(y, A) = limryn =rn n(x) - n(X) dyn(X)

From (6) and (5) we then obtain

(12) K(y, z) = (K(x, z), K(x, y)) = 4 f 4n(Y, X)>f(Z, X)d/nj n -oo

(13) K(y, y) = f 4| n(y, X) 12dn. n -X

The series and the integrals in (12) are absolutely convergent because of (13).

The function cJn(Y, A) is in general defined for each y only almost everywhere in X in the sense of the measure An. Nevertheless, in most applications it turns out to be a continuous function of X. In spite of this, Pn(Y, Xo), as function of y for a fixed N0, will not in general belong to 'D.

11. Operators in spaces with reproducing kernels(9). In a class F forming a Hilbert space with a r.k. K, the bounded operators admit of an interesting representation.

The notation L$K(x, y) indicates that the operator is applied to K(x, y) as function of x and that the resulting function is considered as function of x (but it will depend also on y which will act in the transformation as a parameter). It is then clear what is meant by L.K(x, z), LxL,'K(x, z), and so on. The notation Lf (x) is clear and we may also write Lf (xo) if xo is a particular value of x. Consider the adjoint operator L* (that is, the operator for which (Lf, g) - (f, L*g)). Take the transform

(9) The developments of this section are closely related with the work and ideas of E. H. Moore.




(1) A(x, y) = L.K(x, y).

A is a function of the two points x, y. As function of x it belongs to F. Take then for anyfEF the scalar product (f(x), A(x, y))x= (f(x), Lx*K(x, y))x =(Lf (x), K (x, y)) x= Lf (y),

(2) Lf(y) = (f(x), A(x, y)).

In this way, to each bounded operator there corresponds a kernel A(x, y) which for every y, as function of x, belongs to F. The operator is represented in terms of the kernel by formula (2).

Let us now find the kernel A*(x, y) corresponding to the adjoint operator L*. We have (L*)*=L and thus

(LxK(x, z), K(x, y)) = (K(x, z), LxK(x,y)),

(A*(x, z), K(x, y)) =(A(x, y), K(x, z)),

(3) A*(y, z) = A(z, y).

It is clear that to L1+L2 or acL correspond A1+A2 and &A respectively. We shall now find the kernel A corresponding to the composition L=LIL2. Since (L1 L2) * L= *, we have

A(y, z) = (LjL2)tK(y, z) = L*2LlvK(Y,z) = L27,Ai(y, z) = (Ai(x, z), A*(X, y)) = (Al(x, z), A2(v, x))

(4) A(y, z) = (Al(x, z), A2(y, x)) for L = L1L2.

Let us note the following properties resulting immediately from (1)-(4):

(5) ((f(x), A(x, y))x, g(y)), = (f(x), (g(y), A(x, y))y)x

= (f(x), (A (X, y), g(y))y)x.

(6) The symmetry of L is equivalent to the hermitian symmetry of A:

A(x, y) = A(y, x).

We prove now the following property:

(7) The operator L is positive if and only if A is a p. matrix.

In fact, L positive means that for every fC F, (Lf, f) _O. For f= nZkK(x, yk) we then get

, j TD (L xK(x1 yi), K(x, y j))

- Z Z PiPj(A*(x, yi), K(x, yj)) = A A*(yj, yi)?ivj

- Z Z A(yj, yj)riTi > 0.




This proves that A and thus A is also a p. matrix. It also proves the well known fact that a positive operator is always symmetric.

If now A is a p. matrix, we see that (Lf, f) 0 will be satisfied for all f of the form Z 1K(x, yk). As these functions form a dense set in F, every function fE F may be approximated by them and we get (Lf, f) >0 by a passage to the limit.

In generalizing the notation of ?7 we shall write A1<<A2 or A2<<A1 for any two kernels if A2-A1 is a p. matrix.

THEOREM I. For an arbitrary kernel A(x, y), hermitian symmetric (that is, A (x, y) =A(y, x)), the necessary and sufficient condition that it correspond to a bounded symmetric operator with lower bound _m> - oo and upper bound < M < + oo is that mK<A?MK.

Proof. Necessity. If L is the corresponding symmetric operator with bounds not less than m and not greater than M, we have

m(f, f) < (Lf, f) < M(f, f) for every f C F.

It follows that ((L-mI)f, f) >0 and ((MI-L)f, f) >0, that is, the operators L-mI and MI-L are positive. Therefore, from (7) we obtain that A -mK and MK -A are p. matrices.

Sufjiciency. The condition of the theorem is clearly equivalent to

1 0<< -(A-mK)<<K.

M-m

This means that the kernel K1 = (1/(M -m)(A- mK) is a p. matrix and is <<K. Therefore it is a reproducing kernel of a class F1 with the norm || ill and following Theorem I, ?7, F1 C F and If,flflfj > for f1 X F1. Then, as in Theorem III, ?7, the operator

Lif(y) = (f(x), K1(x, y))

is a positive operator in F with a bound not greater than 1, that is,

O (L if, f) < (f, f).

This operator corresponds to Ki(x, y) by its definition. Consequently, to A= (M-m)Ki+mK there corresponds the operator L= (Mr-m)Ll+mI and the last inequalities give

m(f, f) < (mif, f) + ((M - m)L1f, f) _ M(f, f), m (f, f) < (Lf, f) < M (f, f).-

An arbitrary kernel A is representable in a unique way in the form

(8) A = A1 + iA2, A1 and A2 hermitian symmetric.

Namely, we have




A 1(x, y) - (A(x, y) + A (y, x)),

(9) 1 A2(x, y) =- (A(x, y) - A(y, x)).

The necessary and sufficient condition in order that A correspond to a bounded operator is, clearly, that A1 and A2 correspond to such operators. To the last kernels we can apply Theorem I.

We now consider convergence of operators. The three simplest notions of limit for bounded operators are the following: the weak limit, w. limn.," Ln = L, if Lnu converges weakly to Lu for every u ? F; the strong limit, str. lim Ln = L, if Lnu converges strongly to Lu; the uniform limit, un. lim Ln=L, if IILn-Li| ->0, where || || for operators denotes their bound.

It is clear that weak convergence follows from strong convergence and that strong convergence follows from the uniform one.

It is known that w. lim Ln=L involves the boundedness of all ||L,|| and the inequality I I LI I < lim inf. I I Ln| I .

THEOREM II. If L = w. lim Ln, then for the corresponding kernels we have A(x, y) =lim Alt(x, y) for every x, y in E. If L=un. lim Ln, then An converges uniformly to A in every set of couples (x, y) for which K(x, x) and K(y, y) are uniformly bounded.

The first part follows immediately, by the definition of weak convergence from

A(x, y) = (A(z, y), K(z, x))z = (LzK(z, y), K(z, x))z = (K(z, y), LzK(z, y))z = lim (K(z, y), LnzK(z, x))z = lim An(X, y).

The second part follows easily from

A (x, y) - An(X, y) = I (K(z, y), (L - Ln)zK(z, x))

_ ||K(z, y)IfzlI(L- Ln)zK(z, x)|lz < |IK(z, y)IKzI|L -L11 || K(z, x)I1

- |L- Ln|l(K(x, x) K(y, y))112.

Consider now two orthonormal complete systems in F, {g ', } and { g II} (in particular we may have g' =g"). The double system {Igt (x)g"(y) } is a complete orthonormal system in the direct product F0F.

If A(x, y) belongs to the direct product we know that it is representable by an absolutely convergent double series

(10) A(x, y) = mn9M(X)gn (y) m,n




where the coefficients aemn satisfy E.,, I amn 2< oo and are given by

(11) =mn = (gn (y), (g(x), A(x, y))X)y = (gn$(y), Lg' (y)). THEOREM III. For every bounded operator L, the series in (10) with coeffi-

cients given by (11) is convergent for every x and y in the sense P Iq

(12) A(x, y) = lrn z X xmngm(x)gn" (y) P,q=oo m=l n=1

The A(x, y) belonging to the direct product F0 F correspond to operators with finite norm.

Let P' and Pq" be the projections on the subspaces generated by g and gl', g2, I , gq . It is clear that str. limp=O P =1,

str. limq., P' =I. Consequently, for any u, v in F

(13) rim (P,'v, LP,u) = (v, Lu). P,Q=oo

If we take now v=K(z, y) and u=K(z, x) as functions of z, we get Pq'v -PI'$K(z, y)= Zfg " (z)g,'(y), J%,u= ,:g' (z)g,'(x) and (11) and (13) then lead directly to (12).

The norm of an operator L is given by 1(L)= .E=1=fLgmII[2 for any orthonormal complete system {gn}. It is independent of the choice of this system and may be finite or infinite. From (11) it is clear that

00

Lgm(y) = Eamgn'(y) n==1

by development in the system { ga" }. Consequently, IILg/'(y) 112 = n2

and 91(L) = EL, y,=l | O mnI 2 which proves the second part of our theorem. 12. The reproducing kernel of a sum of two closed subspaces. Let F be a

class with a r.k. K. We know that the r.k.'s of closed subspaces of F correspond to the projections on these subspaces.

The problem of expressing the r.k. of the sum F1EDF2 of two closed subspaces in terms of the r.k.'s K1 and K2 of these subspaces is therefore reduced to the problem of expressing the projection P on F1 DF2 in terms of the projections Pi and P2 on F1 and F2.

In order to obtain this we shall at first prove the identity

[(P - P1) (P -P2) ] (l) ~m

p - E P1(P2Pl) k-1 + P2(P1P2) k-1 - (P2P,) k - (P1P2)k] - (P2P1)m- k=1

We shall use the known properties of projections, namely: P1 = P,P = PP1 -P2, P2 = P2P = PP2 = P2,




P1(P-P1) = (P-PO)P=O0, P2(P-P2) = (P-P2)P2=O.

Then, denoting the expression (P-P1)(P P2) by Q w-e have

Qkpl = Qk-l(p - Pl)(P - P2)Pl = Qk l(P - Pl)(P- P2P1) = - Q'-lP2Pl + Q k-lplP2Pl

If k>1, the first term is _Qk-2(P_P])(P-P2)P2P1=O, and we have

Qkpl = Qk-lPlP2Pl, k> >.

For k = 1 we obtain

QP1 = -P2P1 + PlP2Pl. Finally, we obtain by induction

(2) Qkp1 = (P2P1)k + Pl(P2Pl) k.

Further, we have

(3) Q= (P-P1)(P-P2) = P-P1-P2 + P1P2.

This gives

Q = -(P P1 - P2 + P1P2) = Qn lp - Qn-l Qn-1P2 + Qn-lPlP2

Since Qn-lp = Qn-l, Q(-lP2 =0, we get from (2)

Qn = Qn-1 - Qn-lPl + Qn'lp1P2

= (n-1 - [-(P2P1) n-1 + P1(P2P1) n-1

+ [-(P2P1)n-1P2 + Pl(P2P1) n-1P2

= Qn-1 + (P2P1) n-1 - Pl(P2Pi) n-1 - P2(P1P2) nl1 + (P1P2) n.

This expression is valid for n > 2. Adding these equations side by side for n=m, m-1, * , 2, and using the formula for Q given in (3), we obtain the required formula for Qm. This formula may be written in the form

p = [(P - P1) (P - P2)]m + (P2P1)m

(4) m + E [Pl(P2P1) k-1 + P2(P1P2) k-1 - (P2Pl) k - (P1P2) k].

k=l

We shall prove now that for m- >oo, (P2Pl)m converges strongly to the projection Po on the intersection Fo of F1 and F2. In order to do so we shall consider the operator L P2P1 in the space F2.

In this space L is a positive operator (and therefore symmetric) with bound not greater than 1. In fact, for uCF2

(P2PlU P2Plu) = IP2P1UI 2 I P1ul 2= (Plu, Ply) = (Plu, DU) = (Plu, P2u)

= (P2Plu, U) I lPiu1 2 ' 9u 2




(5) 0 < (Lu, Lu) < (Lu, u) < (u, u).

Consider now for any f E F the sequence { Lkf}. For k > 1, LVf = P2PiLk-lf CF2. Putting u=Lkf in (5), we get

o < (Lk+lf, Lk+lf) < (Lk+lf, Lkf) ? (Lkf, Lkf),

o < (L2k+lf, Lf) < (L2kf, Lf) < (L2k-lf, Lf).

Consequently the sequence { Lnf, Lf) } is a decreasing sequence of positive numbers and therefore it is convergent. This gives

lim JILmf-Lfll2 = lim [(Lmf, Lmf) - (Lmf, Lf) m,n = xo m,n= ~~~ - (Lnf, Lmf) + (Lnf, Lnf)]

= lim [(L2 m-lf, Lf) - 2 (L m+n-lf, Lf) + (L2n-lf Uf) = 0

and thus Lmf converges strongly. This means that Lm converges strongly to some bounded operator Po. We have further Lm+lf=LLmf =LmLf, which, for

m-o 0, gives

(6) LPof = Pof = PoLf.

Therefore LmPof = Pof and P2f =lim LmPof = Pof. In the subspace F2, Po as a limit of symmetric operators is symmetric. Together with Pof=Pof it

shows that in F2 the operator Po is a projection. It is the projection on the sub-

space of all Pof. From (6) we get I P2P1Pof I IIPlPfl < IIPofl = I P2P1Pof| .

Consequently I P2P1PofJJ = JPPof = ||Pof| , P2P1Pof = PiPof = Pof and

Pof C Fo = F1 F2. Inversely if u E F1 F2, then Lu = P2P1u = u and Pou

-lim Lnu=U. Thus, in F2, Po is the projection on Fo. Then for any f CF, we have by (6)

Pof = PoLf = PoP2Plf = projection of f on Fo.

In our formula (4), besides the series : and the term (P2P1)m we have still

the expression (P-P1) (P-P2)m. P-P1 is the projection on F'9Fl, and PeP2

is the projection onF'9F2, if we denote F1 0 F2 by F'. Consequently for m-* o,

the last expression converges strongly to the projection on the intersection

of (F'eF1) and (F'eF2). But this intersection is reduced to the element

zero because were there in it any element u 0, it would belong to F' and

would be orthogonal to F1 as well as to F2. Thus, u would be orthogonal to

F1 0 F2 = F' which is impossible.

In this way we finally obtain the desired formula for the projection P:

00

(7) P = P0 + ,j [P1(P2P1) '-1 + P2(P1P2) k-1 - (P2P1) k - (P1P2) ]. k=l

The subspace F1 0F2 is defined as the closure of the subspace F1+F2 composed of all sums f1+f2, f C Fl, f2C F2. In general F1+F2 is not a closed

subspace.




Formula (7) is especially advantageous when F1+F2 is closed and thus equal to F'=F1ED F2.

Let us analyze this case more in detail. It will be convenient to make the non-essential assumption that

(8) Fo = F1F2 = (0), that is, Po = 0.

The angle between two elements (vectors) fi-0, f250 is given by cos a =Re (fi, f2)/||f, I IIf2I |. The minimal angle X, 0_< 7r/2, between F1 and F2 is given by('0)

(9) cos4? = l.u.b. Re (fl - ) r 0 5 f, C Fl, O 5 f2E F2 (9) 1 lfll I If4Hf2II for fSF,0f2F.

It is easily seen that, for f1 C F,, f2E F2,

(10) I (fl, f2) I _ Ilf4l lIf211 COS X, (11) IIP,f2II ? flf2Il cos4, ?pIP2f1I I Ilfll I cos4, (12) | 1 fl+ f2H ? lifill sin X If,f' + f2 ? I I f2I| sin 0.

In (12), sin 4 is the greatest constant c>O for which an inequality of type ILf,1+f2I > cI|f,I is true. By a theorem of H. Kober [1] such inequality with c>O is necessary and sufficient in order that F1-+-F2 be closed.

Consequently, we shall know that F'= F1 + F2 if we prove an inequality

(13) llfl + f21 _? cllflll, with any c> 0. Such a constant is necessarily less than or equal to 1 and it gives always an evaluation of the minimal angle 0:

(14) sin 4> c> 0.

The angle 0 being positive, the inequalities (11) show that the bounds of the operators (P2P])n in F2 or (PlP2)n in F, are not greater than cos2no.Formula (7) may now be written in the form

P = (Pl - PlP2 + P1P2P1 - P1P2P1P2 +*)

(15) + (P2 - P2P1 + P2P1P2 - P2P1P2P1 +*)

and the two series are uniformly convergent to the operators Q, and Q2 which give the decomposition of f C F1+ F2 in f = Qlf + Q2f, Qlf C F1, Q2f C F2.

It should be remarked that the decomposition of the series in (7) into the two series (15) is not possible when 4 = 0, as the operators Q, and Q2 are then unbounded.

When the series in (15) are used for computation it is very easy to get

(10) The notion of a minimal angle between two subspaces seems to have been first introduced by K. Friedrichs [1]1.




estimates for the remainder. Usually we shall want to compute, for f and g in F, the value of (f, Pg) = (f, Qlg) +(f, Q2g). It is clear that when we stop in the series of Qi at the nth term p(n) -(1)n-1P1P2P1 * then the remainder R") of this series will be given by

(n) (n) (n) (n) (16) R1 = -P1 Q2 for odd n, Rn -P Qi for even n.

We have similar developments for the second series defining Q2. Conse- quently, the error in (f, Qlg) (for example when n is odd) is given by (f, Ri4g) = -(p~)1 *f, Q2g)

(17) 1 (f, Ri>g) ? < p(f)*fJ JJQ2gJJ.

By (12) we have ||Q2g| <(1/sin 0)|1gll. As p(n) is already computed, p(n)* is known also and we can compute IPl(n)*fl . This will give quite a pre- cise evaluation of the error. Without knowing p(n) we can evaluate !Pil)* -I|ffl cOsn?.

Even in case 0 = 0 we could still evaluate the error in (f, Pg) if Qig and Q2g exist and if we can evaluate their norms.

Still another evaluation of error (preferable as an a priori evaluation), in the case 4 > 0, is obtained directly from (4):

m E [ ] =[(P Pi) (P P2) ] m + (P2P1) m- k=1

It can be proved that the minimal angle of F'&F1 and F'&F2 is the same as between F1 and F2. Consequently

|| [(P - P1)(P - P2)]mll < COS2m-1 (j 11 (P2P1) ml| < COS2m-1 4),

p - El[ ] 2< 2mcos2m 4, k==1

where || signifies bounds of operators. In case of a sum of more than two subspaces F'= F1 0 F2D eF3 e * * we

can still express the projection on F' in terms of projections on F1, F2, but the formula will be much more complicated than in the case of two subspaces and for this reason may not be as valuable.

Let us consider now the translation of our formulas in terms of the reproducing kernels K1, K2, and K' of the classes F1, F2, and F'-= F1 0 F2. We shall suppose that the classes F1 and F2 have no function O0 in common, that is, F1 F2= ().

To the projections P1, P2, P there correspond (in the sense of ?11) the kernels K1, K2 and K'. To each term in the series (7) or (15) there corresponds a kernel given by the following table of correspondence

P +- K'x y), Pi A- Ki.\ yn P2 <- K2X Y),zAS




PlP2 <-> Al(x, y) = (Kl(z, y), K2(Z, x)) ,

P2P1 *-> Al(y, x) = (K2(Z, y), Kl(z, x))z,

(Pl1P2) An + .( x, y), (PP1 +> n^

where

An(X, y) = (Al(z, y), An_l(X, z)), = (An1(z, y), An2(x, z))Z, nl + n2 -

Pl(P2P 1) n<-> A' (x, y) = A' (y, x) = (K1(z, y), An(z, x))z,

P2(P1P2)n +-> An (x, y) = A" (y, x) = (K2(Z, y), An(x, z))z.

Formula (15) can now be written in the form 00

K'(x, y) = (A2.1(x, y) + A"21(x, y) -An(x, y) -An(y, x))

(18) n-1

= (An-i(x, y) - An(X, Y)) + E (An2(x, y) - An(y, x)). n=l n=l

If we use these series to compute K'(x, y) for given points x and y, we shall represent it in the form

K'(x, y) = (K'(z, y), PzK'(z, x)) = (K (z, y), PzK (z, x))

and apply our evaluation of error to this form. 13. Final remarks in the general theory. In the present section we

shall collect a number of shorter remarks about the nature of classes of functions with reproducing kernels and of their norms, and concerning some relations between the classes and their r.k.'s.

(A) Classes of functions for which a r.k. exists. (R.K.)-classes. Consider a set E and a linear class F of functions defined (and finite) everywhere in E. The problem which arises is to find under what circumstances we can define a norm in F giving to F the structure of a Hilbert space with a r.k. For abbreviation we shall call such classes of functions (R.K.)-classes.

THEOREM I. In order that the class F (not necessarily linear) be contained in a (R.K.)-class it is necessary that there exist an increasing sequence of sets ElCE2C * * *, E=El+E2+ , * * , and for each f# O of F a positive number N(f), so that the functions f(x)/N(f) be uniformly bounded in each En.

In fact if F1 is the (R.K.)-class containing F, ff f1' and K1 its norm and kernel, we define as En the set of all yCE with Ki(y, y) <n and as N(f) the norm J|fJJl. Then, for each yCEn and fCFCF1 we have lf(y)I = I (f(x), Ki1(x, y))1| ? IfflIIIK,(x, y)IlI = N(f)((K'(y, y))l/2 and If(y) /N(f)<nl12.

The necessary condition of Theorem I is not always satisfied even for an enumerable sequence of functions. As an example, consider in the interval E= (0, 1) the sequence of functionsfn(x) = 1/| x-rnl for x5# rn and fn(rn) =0-




Here { r, } is a sequence of numbers everywhere dense in E. By an easy topological argument we prove that the sequence of functions fn does not satisfy the condition of Theorem I.

THEOREM II. For an enumerable sequence {fn(x) } the condition of Theorem I is also sufficient in order that this sequence be contained in a (R.K.)-class.

In fact, consider an upper bound Mm < co for Ifn(x) I /N(f), n = 1, 2, * x CEm. We write

1 K(x, y) -A fn (X)f4(y)

n= 2 AJnN(f)

Clearly, the series is absolutely convergent and represents a p. matrix. Each term of it

Kn (x, Y) = 2n zN(fn) fn(X)fn(y)

is also a p. matrix and Kn<<K. Theorem I of ?7 gives then for the corresponding classes FnCF. Obviously Fn is the one-dimensional class generated by fn. Therefore fn C F and {fn } C F.

The condition of Theorem I is certainly not sufficient in general. This may be shown by a simple set-theoretical argument. Let us consider namely the class Fb of all bounded functions on E. We can then take En = E, N(f) = l.u.b. If(x) I . If X is the power of E then the power of a (R.K.)-class F is M"o (as the functions K(x, y) =h,(x) form a complete system in F). On the other hand, the power of the class Fb is -C0 (c is the power of continuum) and for > N, c0>A o.

(B) Convergence in classes with reproducing kernels. Consider a class F with a r.k. K. We know that if fn converges strongly to f in F, then it converges uniformly in every subset of E where K(x, x) is uniformly bounded. Therefore the sequence {fn } satisfies the condition

(1) fn(x) ->f(x) for every xCE, the convergence being uniform in every set of an increasing sequence of sets El CE2 C * . - with E = E1 + E2+ .

Consider now the class 4? of all functions defined in E. In 41 we can introduce a notion of limit as follows:

(2) f(x) = .4-lim fn(x), if condition (1) is satisfied.

It is clear that in general the sequence of sets En will depend on the sequence {fn }. We can now formulate the following theorem.

THEOREM III. In every class F with a r.k., the strong convergence of fn(x) to f(x) involves J-lim fn(x) =f(x).

It should be noted that the weak convergence in F does not involve in




general the D-convergence. But there are important cases where even weak convergence involves (D-convergence. Such cases were considered in ?2, (5).

(C) Relations between (R.K.) -classes and corresponding norms and reproducing kernels. To a p. matrix there corresponds a uniquely determined class and norm, but to a (R.K)-class there correspond infinitely many norms giving to it the structure of a Hilbert space with a r.k. Consequently to a (R.K.)-class there correspond also infinitely many p. matrices which are r.k.'s of the class for convenient norms.

If the norm || || corresponds to a (R.K.)-class F, the norm IIi=clI || c> 0, obviously also corresponds to F and the corresponding r.k.'s K and Ki satisfy

Kl(x, y) - K(x, y).

In fact, the scalar product (, ), is clearly =c2(, ) and thus f(y) =(f(x), K(x, y)) =c2(f(x), (1/c2)K(x, y)) = (f(x), (1/c2)K(x, y))'.

We shall now have to apply an important theorem of S. Banach [1 ] in the theory of linear transformations.

Let T be a linear transformation of a linear subspace F' of a complete space F on a linear subspace F1 of a complete space F1. The subspaces F' and F1' are not necessarily closed. The transformation T is called closed if from {f,n } C F', f-*>f G F, and Tf->fi F1 follows f G F', f1EFl' and Tf =fi.

BANACH'S THEOREM. If T is a closed linear transformation of F' on FL, F' C F, F, C F1, F and F1 complete normed vector spaces and if F' is a closed subspace of F, then T is continuous and consequently bounded (that is, there exists a M>0 with I Tffl1 < MI f| ). The image Fil is either = F1 or of first category in F1.

Before we apply this theorem we shall prove the following lemma:

LEMMA. Let F1 and F2 be classes with r.k.'s and let Fo be their intersection F1* F2. The correspondence transforming f G Fo considered as belonging to F1 into f considered as belonging to F2 is a closed linear transformation.

In fact, suppose that {f,, I CFo and that fr converges strongly to f' in F, and to f" in F2. Following Theorem III

f' (X) = Dlim fn (X) =f// (X).O

Therefore, f' =f" C Fo, which proves the lemma.

THEOREM IV. Let F and F1CF be (R.K.)-classes and | , | some norms corresponding to F and F1. Then there exists a constant M>O such that

?MIJf < IMJ|fijforfEFi.

In fact the identical transformation of F1 considered as subspace of F1




on F1 as subspace of F is closed (following the lemma), F1 is a closed subspace of F1, and thus our theorem follows immediately from Banach's theorem.

COROLLARY IV1. Let and 1 be two norms corresponding to the same (R.K.)-class F. There exist two positive constants m and M such that m||fI| -||f||l ? Mffl| ,forfCF.

Theorem IV, together with Theorems I and II from ?7 and with the remark that to norm M|| || corresponds the kernel (1/M2)K, gives immediately the corollaries:

COROLLARY IV2. Let K and K1 be two p. matrices, F and F1 the corresponding classes. In order that F1 CF it is necessary and sufficient that there exists a positive constant M such that K1<<MK.

COROLLARY IV3. Under the hypotheses of corollary IV2, in order that F1 = F it is necessary and sufficient that there exist two positive constants m and iV[ such that mK<<K1<<MK.

The second part of Banach's theorem together with our lemma leads to the following remark which belongs to the subject matter of section (A).

REMARK. If { F. } is a strictly increasing sequence of (R.K.)-classes, then their sum F= EFn is not a (R.K.)-class. In fact, were there a norm || ||

in F giving it the structure of a Hilbert space with r.k., the subspaces FnCF would be of first category in F and therefore F would be of first category in itself which is impossible.

(D) Connection with existence domains in a Hilbert space. We shall now use the notion introduced recently by J. Dixmier [1 ] of domains of existence in a Hilbert space. A linear subset of a Hilbert space is called a domain of existence, d.e., if there exists a closed linear transformation defined in this subspace and transforming it into a subspace of another Hilbert space (which, in particular, may be identical to the first Hilbert space). The d.e.'s D in a given Hilbert space & with norm may be characterized by the following property: there exists a norm | defined in D giving it the structure of a Hilbert space and satisfying

(3) fllzfl _ for every h G D.

In fact, if D is a d.e., then we consider the linear closed transformation T of D into a subspace of some Hilbert space t', with the norm || P. It is then clear that the norm | f| defined by

||h =12 1 h112 + .. 2

gives D the character of a complete Hilbert space which satisfies condition (3).




On the other hand, suppose that a norm || 11, is defined in D satisfying (3) and giving D the character of a complete Hilbert space. Then the correspondence transforming any element of D, considered as a subspace of &, into the same element considered in the Hilbert space D (with norm is obviously a closed transformation and D is therefore a d.e.

Using Theorem II from ?7 and Theorem IV of section (C) we prove now immediately the following theorem.

THEOREM V. If a class of functions Fforms a Hilbert space with a reproducing kernel, then for any linear subclass F1 C F, the necessary and sufficient condition in order that F1 be a (R.K.)-class is that F1 be a d.e. in F.

If we have two classes of functions F,, F2, with reproducing kernels, we can combine these two classes in different ways in order to form new classes. Let us consider in particular the following classes of functions: Fo=Fl F2 and F==-F+F2.

THEOREM VI. If F, and F2 are (R.K.)-classes, then the same is true of the classes F,* F2 and F1+ F2.

Proof. The linearity of the classes is obvious. We take firstly the intersection Fo. With any norms || |1, and || 112 corresponding to F1 and F2 we define the norm in Fo by the equation

=2 = ffll2 + 11f112 This norm clearly defines a quadratic metric in Fo satisfying all the re-

quired properties. For instance, the completeness of Fo results immediately from the lemma of section (C).

As I |f| I > I f I Ifi for f E F0, Theorem I I, ? 7 gives then the existence of a r.k. for Fo.

In the case of the sum, F= F1+ F2, we may apply the theorem of ?6 which states that K1 and K2 being the r.k.'s with the norms || |1, and || 112 of F, and F2, K1+K2 is the reproducing kernel of our class F.

Besides the operations of and +, we can also introduce the direct product F1? F2 as defined in ?8 as another operation leading to a (R.K.)-class when F1 and F2 are (R.K.)-classes. The class F1?, F2 however is defined not in E but in the product set EXE. If we take its restriction to the diagonal set of all pairs { x, x }, we get a class of functions defined in E. It can be proved that this class does not depend on the choice of norms in F1 and F2 as long as the norms give to F, and F2 the structure of a Hilbert space with a r.k.

PART II. EXAMPLES

1. Introductory remarks. In this part we shall give examples showing how our general theory may be applied in particular cases and to what kind of results it leads. With a few exceptions we will not go into the details of calcula-




tion and will not give in explicit form the formulas and relations obtainable by our general methods.

We shall treat essentially two kinds of kernels: the Bergman's kernels K(z, z1) and the harmonic kernels H(z, z1).

(1) Bergman's kernels. These kernels correspond to a domain D in the space of n complex variables z= (z(1), z() Z * (n)). We consider the class 21 =fD of all analytic regular functions in D with a finite norm given by

lifil2

= J,J. f(z(') Z(2), Z ..(n) 12dx(1)dy(1)dx(2)dy(2) * dx(n)dy(n),

where z(k) = x(k) +iy(k).

The class 2 possesses a reproducing kernel K--KD-the Bergman's kernel corresponding to D.

In our examples we shall consider essentially the case of plane domains D. If D is multiply-connected we shall consider also the reduced Bergman's kernel K'(z, z1), which is the reproducing kernel of the subspace 2' of 2f consisting of all functions of 2I with a uniform integral fzfdz. If D is of finite connection n, the complementary subspace seaf' is (n-1)-dimensional and is generated by n functions w ' (z) (between which there is the linear relation

Wk =0) defined in the following way: if Bk, k=1, 2, * , n, are the boundary components of the boundary B of D, w' is the derivative (which is uniform) of the multiform analytic function Wk whose real part is the harmonic measure uk of D corresponding to Bk, that is, the harmonic function regular in D, equal to 1 on Bk, and vanishing on all the other components Bi.

The functions wk belong always to 2I and are orthogonal to 2'. We have the relation

K(z, zi) = K'(z, zi) + Z ciiw (z) w (zi), i, j

where E is the r.k. of W.&W. Consequently the matrix { cii } is definite positive (see ?3) and it is the conjugate inverse of the Gramm's matrix { (wI', wf) }.

Bergman's kernels possess an important property of invariance: in case of domains in the space of n variables z(1), , (), if T represents D pseudo- conformally on D', then

KD'(z', zL)aT(z)aT(z1) = KD(Z, Zl).

Here, z'= T(z), z1' = T(zi), and AT(z) is the Jacobi determinant of T. In the case of domains in the plane, if t(z) represents D conformally on D', this formula takes the form

KD(Z', Z1) t'(Z) t(z 1) = KD(z, z1).




The importance of the Bergman kernels lies in the possibility they offer of generalizing different theorems on analytic functions of one complex variable to functions of several complex variables (such as Schwarz's lemma, distor- tion theorems, representative domains in pseudo-conformal mappings).

In the case of one variable almost all the important conformal mappings are expressible in terms of these kernels. For instance if D is a simply connected domain, the mapping function P =f(z, z0) which represents D on a circle I| <R insuch away that the point zoED goes into =Oandf'(zo, zo)=1 is given by

1 z f(z, zo) = K(zo, zo) K(t, zo)dt.

(2) Harmonic kernels. Consider in a plane domain D (we could consider also a domain in n-dimensional space) the class e-3-D of all regular harmonic functions (in general complex-valued) with a finite norm given by

= f k|2dxdy, z = x + iy.

This class possesses a reproducing kernel which will be denoted by H(z, z1).

It should be remarked that another harmonic kernel is often considered, namely the one which corresponds to the Dirichlet metric

||k|2= [I |( 12 + I 1 |12]dxdy.

This kernel is easily expressible by Bergman's kernel and consequently does not present any additional difficulties to the ones encountered in the study and computation of Bergman's kernels.

The situation is different for the kernel H. Even for very simple domains (for instance for a rectangle) there is no known explicit expression of H even in the form of an infinite development. (We disregard here the developments in terms of a complete non-orthogonal system which are always possible to establish for a r.k., but whose coefficients are quotients of determinants of growing orders.)

One reason for the greater difficulty of the investigation of the kernels H(z, z1) as compared to Bergman's kernels lies in the fact that H has no such invariancy property vis-a-vis conformal transformations as have Bergman's kernels.

The interest of the kernel H lies in its connection with the biharmonic problem which governs the question of equilibrium of elastic plates.

The kernel H gives a simple expression for a function u(z) such that u = au/an =0 on the boundary B of D and AAu = f in D, for a given function q5.




Supposing that we know a function 61 such that A4I = ) (we can take as V/ the logarithmic potential of 4: 4'(z) = (1l7r)ffD log Iz-z'J 4(z')dx'dy') we get for u the expression (where g is the ordinary Green's function)

u(z) = ff g(z z')dx'dy' [(z') - H(z', z">1)q(z")dx"dy"].

The Green's function gIr(z, z1) of the biharmonic problem, satisfying AAgII= O for zOz1, gII=OagII/n=0 on the boundary, is given by

gII(z, z1) = g(z, z')g(z'. zl)dx'dy'

- t ff g(z, z')dx'dy' ff H(z', Z")g(z", zj)dx"'dy "'.

These formulas were essentially noticed already by S. Zaremba [2]. 2. Comparison domains. Consider two domains in the plane, D and

D', DCD'. The kernels KD' or HD,, restricted to the domain D, are reproducing kernels of classes W? or Q3 formed by the restrictions of functions from ED' or 3D'. As any analytic or harmonic function vanishing in D vanishes everywhere, any function fo of W? or V3 is a restriction of only one function f from 2fD' or E3D, and, following ?5, Part I, the norm I|fo1Io| 0=fI'. It is then clear that every fo ?2V belongs to ED and that I|fo 0 o>IfoIl.

We can apply Theorem II of ?7, I, which gives

(1) K ?, << KD, KD' being the restriction of KD' to D.

In the same way we get 0

(2) HD' << HD.

If the kernel KD is known, we get immediately the well known estimates for the kernel KD':

(3) KD'(z, z) < KD(Z, z), IKDI(Z, Z1) I < (KD(z, z)KD(z1, Z1)) 12

for points z and z1 belonging to D. But the relation (1) (or (2)) allows much better estimates. Suppose that

the kernel KD' is known. For two points z and zi in D take domains D" and D1" such that zED"CD, z1EEDP CD and that the kernels KD" and KD1' be known (for instance circles). Then, from (1), we get KD(z, z) <KD;'(Z, Z), KD (ZI, z1) _ KD1, (z1, z1),

KD(Z, Z1) - KD' (z, Z1) 2

(4) < [KD(z, z) - KD'(Z, z) ] [KD(z1, z1) - KD'(z1, z1)] ? [KD"(z, z) - KD'(z, z)] [KD1'(Z1, Zi) -KD'(Z1, Zi)].




If we consider a boundary-point t where the boundary has a finite curva- ture and if we fix z1 and move z towards t, the estimate (3) will grow like 1/I z-t . The estimate (4) by a convenient choice of the comparison domains D', D", and D' will give a bound for I KD(Z, Z1) I growing only like 1/ z-tl 1/2.

To show the interest of this improvement, take D simply-connected and consider the conformal mapping of D on a circle | <R given by

1 frz I K(z, zo)dz.

K(zo, zo) J 0

Our problem will be to compute the point r on the circumference | R corresponding to t on the boundary B. As the kernel K is not known we approximate it by a development in orthogonal functions. This development may converge fairly quickly inside the domain but in general it will not converge on the boundary and will converge less and less well the nearer we come to the boundary.

To calculate r we have to integrate from z0 to the point t on the boundary. We cannot integrate term by term the development of K as it does not converge on the boundary. What we do then is to find, with the help of the estimate (4), a point z1 near t for which the integral fAtl K(z, z0) I dz is sufficiently small. We can integrate the development of K term by term from zo to z1 and obtain as good an approximation of r as we wish.

It is clear that with the estimate (3) we would not be able to do this. 3. The difference of kernels. As we saw in ?2, in DCD1, the kernels

K-KD and K1 KD, satisfy the relation K1<<K (the kernel K1 being restricted to D). To illustrate the developments of ?7, I, let us investigate the class of functions F2 corresponding to the p. matrix K2 given by

(1) K2(z, Z1) = K(z, z1) - K1(z, z1) z and z1 in D. Following the notation in the proof of Theorem II, ?7, I (where F=W,

F1 = ), we introduce the operator L in af by

(2) Lf = fl(z1) = (f, K1(z, z1)) = f f(z)Kl(z, zl)dxdy.

If we consider the Hilbert space 11 of all functions u(z) in square integrable in D1 with the norm

||u||1 = I u u(z) I2dxdy,

the general property of r.k.'s as projections shows that f1(z), as function in D1, is the projection on 2f1 of the function f(z) =f(z) in D and =0 in D1 - D. Consequently,

IIfi(z) ? I fdz)Ill ? Wf(z)1 = Ilf(z)fl.




The second inequality may become equality for f(z) 50 only in the case when D1l-D is of two-dimensional measure 0 (for instance when D differs from D1 only by some slits). We will exclude this case and consequently

(3) 1{fi(z)II = IILflI < WIfil for f - O

We introduce then the operator L' by

= I - L.

The subspace Fo is here reduced to 0 as ff-= Lf is impossible in view of (3). Therefore F'=F--% and the only possibilities for the class F2 are: 1, F2 =9 or 20, F2 is a dense subspace of W.

The first case represents itself always when D is completely interior to D1 (DCD,). In fact, the operator L is then completely continuous. To prove this we take a sequence { g(n) } C2I converging weakly to gE?. The functions g(n) converge then weakly in 11 to g and their projections g (n) on 2[i converge weakly to gi. But the weak convergence in W, involves uniform convergence of g(n)(z) towards g1(z) in any closed subset of D1, in particular in D (see section (5), ?2, I). When we restrict the functions g('(z) and gl(z) to D they become the transforms Lgff) and Lg. Therefore, the uniform convergence of gn) to g, in D involves the strong convergence of Lg(n) to Lg in the space 2f.

Following our remarks after Theorem III, ?7, I, the class F2 = F' = W. To get the norm 1|f112 corresponding to the kernel K2, we have to find the solution g(z) of the equation

g - Lg f which exists and is unique for every fCGf. Then

IIJII2 _ ffglj2 - l[gljll where

gl(zl) = Lg = fg(z)Kl(z, zl)dxdy.

Let us note that the operator L which in general has a bound not greater than 1 has here a bound less than 1.

To illustrate the second case we have to take a domain having common boundary points with the boundary of D1. It seems probable that for every such domain D we shall be in the second case (at least if one of the boundary components of D arrives at the boundary of D1).

To prove that we are in the second case, we have to show that the operator L has a bound =1. Then L cannot be completely continuous (as the bound is not attained). The class F1 is a proper subclass of W[. The class F2? of all functions



390 N. ARCNSZAJN [May

f2(Z) = g(z) - gi(Z), where gi = Lg, g ? 2, is a proper subspace of F2, dense in F2. For such a function f2, the norm in F2 is given by

IIf2I!2 = l2g - jlgilli.

The class F2 in the metric of 1 is a dense subspace of W1. There are functions f Ez which do not belong to F2 and for which, a fortiori, the equation f=g-Lg has no solution g I2f. (There may be such a solution, analytic in D but not in square integrable in D.)

For two explicitly given domains D CD1, it may not be easy to prove that we are in the second case by using the property that the bound of L is 1. We can transform this property into another one, more easily proved.

To this effect we shall consider for any function fi, Ez2 the quotient

(4) Q(fi) = _ffI2/fIfjII2

LEMMA. In order that the bound of L be 1 it is necessary and sufficient that there exist functions fi with Q(fi) as near 1 as we wish.

To prove this lemma we remark firstly that the l.u.b. Q(fi) for f1C 1G is the same as the l.u.b. Q(Lf) for fEz2. In fact the Lf form a dense subspace in the space 211 (otherwise there would be a gF2f1, g P?, with (ga, Lf)1=0 for every fEz1. Therefore (gi, f) = (ga, Lf)1 =0 and gi =0 in D which involves g= 0 in D1). Consequently, there is for every fiC1Ez a function fE21 with 1f,1-LfII1 as small as we wish. As 1f,1-LfII <?1f1-Lf I1 it is clear that Q(Lf)

will approach Q(fi) as nearly as we wish. Now, Q(Lf) can be represented as

(5) Q(Lf) (Lf, Lf) (Lf, Lf) (Lf, Lf)1 (Lf, f)

Our lemma amounts to the equivalence of the two properties, for any ae, 0< oa?1:

(6') (Lf, Lf) ? a 2(f, f) for all f ? 21,

(6") (Lf, Lf) < a (Lf, f) for all f C W.

From (6") follows (6'):

(Lf, Lf) < a(Lf, f) a a(Lf, Lf)12(f, f)1/2

(Lf, Lf) 1 /2 < a(f, f)1/2, (Lf, Lf) c 2(f, )

From (6') follows

(6"') (Lf, f) < a (f, f) for all f E 21.




In fact, we have

(Lf, f) < (Lf, Lf)112(f, f)1/2 < a(f,f)112(f, f)12 = a(f, f).

From (6"') follows (6"): we use the fact that L is positive which gives Lf, g) < (Lf, f) "2(Lg, g)1/2* Then

(Lf, Lf) (Lf, f)l12(LLf, Lf)"12 < (Lf, f)/2a112(Lf, Lf)1/2,

(Lf, Lf)12 < a'12(Lf, f)1/2, (Lf, Lf) < ac(Lf, f).

This proves our lemma. We shall apply the lemma to two domains DCD, having a common

boundary point which belongs also to the boundary of the exterior of D. We suppose further that at this boundary point the boundaries of D and Di have a common tangent. We can take the common boundary point as the origin 0 and the inner normal of the boundaries at this point as the positive axis.

It is then easily verified that the functions

1 f (z) =

n n = 1,2,...,

for sufficiently great values of n, belong to W, and that they satisfy the asymptotic equation

IIfnlI ' II|fnII| for n -m.

This shows that the l.u.b. Q(f) = 1 and that we are in the second case. Let us consider now another kind of example which we excluded till now.

Namely, we shall suppose that the domain D differs from Di only by a number of slits of finite length. It is then immediately seen that for a function fi CEz2, considered as belonging to C?1 (s? =class W, restricted to D), we have

IIf'II = flf,jjl. Consequently 2fV is a closed linear subspace of 2 and the kernel K(z, z1)

- Kl(z, z1) corresponds to the subspace WOWG?. 4. The square of a kernel introduced by Szego. We shall now give an

application of Theorem II, ?8, I. Consider a domain in the plane with a sufficiently smooth boundary (for simplicity's sake we may suppose that the boundary curves are analytic). For this domain we shall consider a kernel, first introduced by Szego [1], which we shall denote by k(z, z,). This kernel corresponds to the class S of all analytic functions which possess in square integrable boundary values. As the functions are not necessarily continuous on the boundary we have to specify the meaning of the boundary values of the functions. We shall suppose that for such a function f(z), its integral F(z) is a continuous function in the closed domain (but F(z) may be a multi-




form function). Then we suppose that for any two points tl, t2 of the same boundary curve the difference F(t2)- F(t1) may be represented by the integral

rt2 ft2f(t)dt,

where f(t) is defined almost everywhere on the boundary curves and is in square integrable there. The integral is taken over an arc of the boundary curve going from t1 to t2.

The function f(t) will be considered as the boundary value of f(z) in the boundary-point t. f(t) is completely determined by f(z) with exception of a boundary set of linear measure zero. The existence of boundary values of f (z) in our sense is equivalent to the absolute continuity of F(z) on the boundary. With the boundary values taken in this sense we define a norm in class S by the equation

liffl 2 = E f(t) 12ds, cv

where Cv are the boundary curves and ds is the element of length on the curve. For the functions of class S it is easily proved that the Cauchy theorem is still valid in the form

f(z)= j dt. CV t- z

From this it is immediately seen that the class possesses a reproducing kernel which is the kernel k(z, z1) introduced by Szego. Quite recently, in his thesis, P. Garabedian [1 proved that

I (1) k2(Z, Z1) =- K(z, z1) + E ai, jw (z) w (zi)

47r i j

where K is the Bergman kernel for the domain and wj' are the functions introduced in ?1.

In cases of simply-connected domains Garabedian's formula takes a very simple form, namely:

k2(z, Z1) -K(z, z1). 47r

In this case Theorem II, ?8, I, gives a property of analytic functions which seems to have been unnoticed even for this simple case. Every function f(z) in square integrable in the domain D is representable in infinitely many ways by a series

(2) f(z) = E ckk(Z)>k(Z)




where the functions k(z) and 41k(z) are in square integrable on the boundary. In addition, we have the formula

(3) 47rf I f(z) 12dxdy = min E 2 f 4k(t)>k(t)ds k(t)Az(t)ds, D k ICC

the minimum being extended to all representations of f(z) in the form (2). In particular, we have the inequality

4 fr j | +(Z) (Z) ?dxdy <f +(t) 12 ds I | (t) j2ds

In the case of a multiply-connected domain the problem is a little more complicated because of the presence of the functions w'. It can be proved then that if we replace the Bergman's kernel K by the reduced kernel K' (see ?1) we have again a formula similar to (1):

(4) k2(Z, Z1) =-K'(z, zi) + >2 fi jw' (z)w! (zi) 47r ti

where, now, the fi, are the coefficients of a positive quadratic hermitian form, which means that Y i, represents a positive matrix. Consequentlv, the functions of the class corresponding to the kernel k2 are sums of functions in square integrable in D with a uniform integral (which form the class belonging to K') and of a linear combination of the w' (which form the class corresponding to 2i).) The functions w' are in square integrable in D, and thus every function belonging to the class of k2 is in square integrable in D. Con- versely, it can be proved that every function in square integrable over D belongs to the class of k2. By Theorem II, ?8, I, we know that the functions belonging to k2 are of the form (2), but we will not be able to obtain a formula like (3) for the case of a simply-connected domain. However, a more complicated formula generalizing (3) does exist. In the present case of a multiply- connected domain, we have still the property that the product O(z)/(z) is in square integrable in D if q and 41 are in square integrable on the boundary, but the inequality between the integrals will not be as simple as in the case of a simply-connected domain.

5. The kernel H(z, zi) for an ellipse. We shall construct the kernel H(z, zi) for the ellipse D

x2 y2 (1) ~~~~~~D: 2 + = ly a > b,

a2 b2

by use of an orthonormal complete system( (i).

(11) The system and the corresponding expression for the kernel were communicated to us by A. Erdelyi. It should be noted that the system was already introduced by S. Zaremba [1 I who also noticed that it is orthogonal and complete in the Dirichlet metric.




We write

(2) a = h cosh e, b= h sinhe,

(3) z= x+ iy=hcoshP, P=t+in.

(3) gives us a conformal transformation of the rectangle R, 0 <t <e,

-<7r <7r on the ellipse (1) with the rectilinear slit -a<x<h. Consider in the rectangle R the analytic function p.(z) = sinh n?/sinh v (positive integer n). It is immediately seen that in the variable z the function is a polynomial of degree n -1. Consequently, all the polynomials in the variable z can be expressed as finite linear combinations of these polynomials for n -1, 2, 3, * . . Since the harmonic polynomials form a complete system in our class e3 of harmonic functions, the real and imaginary parts of pn(z) also form a complete system. On the other hand, it is easy to verify that these real and imaginary parts form an orthogonal system. This verification is made in an easy way by performing the integration in the rectangle R instead of the domain D. Using the formula

J'ff(x, y)dxdy = h2 ff(x, y) I sinh v j2did-

one verifies easily that the sequence q)n(z) defined by

2 11\l2

q52n-2 = (-h ) (sinh 2ne + n sinh 2e)-1/2 Re pn(z), n = 1, 2, *

(4) k2n-3 = -()" (sinh 2ne - n sinh 2e)-1/2 Im pn(z), i = 2, 3, * *

is orthonormal. We can then write the kernel of our class in the form

4 ( Re pn(z) Re pn(Zl) Im pn(Z) Im pn(Zl) (5) H(z, z1) =-2 n -- + i s +

h 7r n=l (sinh 2ne h n sin Z si snh 2ne - n sinh 2e

6. Construction of H(z, z1) for a strip. Our next example will use the theorem of the limit of kernels for decreasing sequences of classes (see Theorem I, ?9, I). It will be at the same time an example of a representation of a kernel by use of a resolution of identity in a Hilbert space (see ?10, I). Consider the kernel of ?1 and suppose that in the ellipse

x2 y2

a2 b2

the axis b <a remains fixed, while a-- oo. The ellipse in the limit will become the horizontal strip Iyf <b. It is clear that our theorem on the limit of kernels applies in this case and we get the kernel H(z, z1) for the strip as a limit of the kernels corresponding to ellipses. Before performing this passage




to the limit, we shall at first make a few preliminary remarks. As in the preceding paragraph, we consider the quantities h and e given by

b = h sinh e, a = h cosh e, b being fixed and a--> co. We see immediately that hc> so and e->O in such a way that he-+b.

In the conformal mapping, z =h cosh v we introduce a new variable ?' by the equation

7r

eD = v- 2 Then z = h sin icr', and the ellipse with the slit along the real axis going

from +h to -a is transformed in the rectangle

3gr 7r R eY 0 < t' < 1, - - < 77 <--

Consider a point z in the horizontal strip IyI <b. For sufficiently large values of h the ellipse will contain z. Suppose that Im z>0, then the corresponding ?' will lie in the upper half of the rectangle Rf and for h-+*oo, ?'-*z/bi. The point T" = D'-xi/c will then correspond to the conjugate point 2, so that (v" +wi/c)->zi/b. If we now return to the formula for the kernel from ?5 and replace the function pn(z) by the expression sinh n?/sinh ?, then replace ? by ci'+iri/2 and separate the series which expresses the kernel H (see (5), ?5) into parts corresponding to even and odd indices, we can write the kernel in the form of a sum of four series:

cos inED' cos inEl1' Re -Re

4 t cos ieD' cos ic-1' 7rh n odd h sinh 2nE+ n sinh 2e

cos inf?' COS infE Im - m -Im

4 n cos iED' cos iED

rh n odd h sinh 2nE - n sinh 2E

sin inED' sin inEP Re -Re

4 n cos iED' cos iE? I _ - - .

7rh n even h sinh 2nE + n sinh 2E

sin inED' sin inEDf Im Im

4 ii COS ift COS i'Ef1

irh n even h sinh 2nE - n sinh 2E

For h-* oo, if we denote n/h by t and if we then notice the asymptotic formulas he-b, nE-bt, n sin 2e-2bt, it is immediately seen that the series represent approximating Riemannian sums for the integrals




2 00 Re cos ibtW' Re cos ibtA; 2 Im cos ibtV' Im cos ibts[ - g

~~~~~~~~t d+ - Xt -dt 7r J sinh 2bt + 2bt 7r sinh 2bt - 2bt

2 rX Re sin ibtW' Re sin ibtP1 2 rX Im sin ibtW' Im sin ibtP; +- t 1dt + - t dt,

7r J 0 sinh 2bt + 2bt 7d + - sinh 2bt - 2bt

when we subdivide the infinite interval (0, + oo) into equal intervals of length 2/h and in each interval take the value of the integrated function in the center (for the first two series) or in the right end (for the two last series).

The convergence of these sums towards the integrals for hI-> co is easily verified and in this way we obtain for the kernel of the horizontal strip the expression given by the above four integrals where we replace ?' and ' by the values z/bi and z1/bi:

2 r0 XRe cos zt Re cos z1t + Re sin zt Re sin z1t H(z, z1) = -tdt

irJo sinh 2bt + 2bt

2 ro Im cos zt Im cos zit + Im sin zt Im sin ztt +- - - tdt.

7r sinh 2bt-2bt

This integral representation of the kernel corresponds to a resolution of identity in the Hilbert space e3 corresponding to the strip. This resolution of identity has a quadruple spectrum (see ?10, I) defined by the following four functions fk(z, X) fk(z, X) = 0 for X < 0, forX > 0

iX sin z 1- cos zX J(z, X) = Re cos ztdt = Re -- f2(z, X) = Re

f3(Z, X) = Im -- , f4(Z, X) = Im z z

It is easily verified that these functions satisfy the conditions (a) and (b) from ?10, I. The condition (c') results from the fact that the functions fk

determine the r.k. H(z, z1) of the class e3 by the formula above. 7. Limits of increasing sequences of kernels. In the preceding section

we had an example of a limit of decreasing sequence of classes and kernels. We shall give here an example of an increasing sequence of kernels.

Consider the Bergman's kernels Kn for a decreasing sequence of simply- connected domains Dn such that Dn+1 CD, and E = lim D. = D, - D2 . . . - con- sists of the two closed circles C1: I z-21 < 1, and C2: ! z+2 1 _ 1, with the seg- mentof thereal axisI: -1<x?<,y=0.

Following the general theory of ?9, I, we have to consider the set Eo where Ko(z, z) = lim Kn(z, z) < ?c*

Every point of the open circle C1 belongs to Eo. In fact, if K(1) is the Berg- man's kernel for C1, by the method of comparison domains (see ?2) we get




Kn<KK(,) in C1 and consequently K(z, z) =lim Kn(z, z) -<K(1)(z, z) for z in C1. The same is true for zE C2. We are going to prove that

(1) Eo = C1 + C2.

We have to show that for zoGE-Eo, lim K5(zo, zo) = oo. We use again a domain of comparison. We take a closed line L defining an exterior domain S containing E and such that by a convenient translation we can approach L as near as we wish to z0 without touching E. The domain S will contain all D, from some n onwards. For these n, K>>Ks, therefore lim Kn(zo, zo) >Ks(zo, zo). The translation of the line L (and the domain S) will have on Ks(zo, z0) the effect as if the domain S were fixed and the point zo were moving towards the boundary L. As for Bergman's kernels of domains of finite connection we have the theorem that K(z, z) goes uniformly to + oo when z ap- proaches the boundary (12), we arrive at the result, lim Kn(zo, zo) - + o.

For zo E-Eo, with exception of z0= ? 1, we can take as L a sufficiently small circumference. For zo = + 1, a circumference cannot do. We take then for L the boundary of a square, for instance, for zo + 1, we take the square: -E<X<1-E,f <y<1+E.

Using the notation of ?9, 1, (B), we see immediately that the class Fo with the norm o is here a subspace of the class F of all functions f(z) defined in the two open circles C; and C2, analytic and regular in each (but f(z) in C1 is not necessarily an analytic continuation of f(z) in C2), with a finite norm

lfl2 f I f(Z) 12dxdy + f I f(z) 12dxdy.

cX C~~~~~~~~2

Consequently, the condition 2? of ?4, I is satisfied and we obtain a functional completion Fo of Fo. Then, it is easily proved, by using general approximation theorems of analytic functions, that the space Fo coincides with F.

The r.k. of F is immediately seen to be given by:

K(z, z1) = 0 if z and z1 belong to different circles,

K(z, zi) = K(i)(z, zi) if z and z1 belong to Ci.

Similar results can be obtained if the domains Dn, Dn?1CDn have an arbitrary intersection E = -D, . . - The set Eo is then the set of all interior points of E.

8. Construction of reproducing kernels by the projection-formula of ?12,

(12) This theorem is proved by using conformal mapping and the invariancy property of Bergman's kernels. For the kernels H for which the invariancy property is not true, a similar theorem has been proved only for domains with sufficiently smooth boundaries.




I. The formula of ?12, giving the r.k. of a sum of two closed subspaces, may serve in many cases for the construction of kernels. We shall indicate here a few cases when it can be applied.

(1) The expression for H(z, z1) in terms of K(z, z1). Consider for a domain D in the plane the classes f and Q3. In spite of the similarity of the metrics in the two classes, the relationship between their kernels K and H does not seem a simple one. We shall get such a relation by using the formula (18), ?12, I.

To this effect we remark firstly that the class Q3 is a class of complex valued functions, that is, every function h of the class is representable in the form h=hi+ih2, hi and h2 harmonic and real-valued. Consequently, the class 9f is a linear closed subspace of the class 3. Also the class a of all antianalytic functions (that is, conjugate f(z) of analytic functions f) with a similar norm is a closed linear subspace of d. On the other hand, it is clear that every function hC,1 is representable as a sum

h(z) = fl(z) + f2(z)

with two analytic functions fi and f2. These functions are uniquely determined with the exception of additive constants. In general, in this decomposition, the functions f, and f2 may be of infinite norm, that is, they may not belong to our classes f and W. But when the boundary of D is not too irregular it may be proved that a dense subclass of e is decomposable in this form with fi and f2 belonging to W and Ft which means that Q3 =W ( W. In order to avoid the indetermination of the above decomposition of htC~3 into fl+f2 because of the additive constant, we shall admit to the class W only functions f satisfying the condition

ffDdxdy = 0.

With W so fixed, the condition W W= (0) is satisfied. Consequently, we can apply our formula to calculate the kernel H.

We obtain for H a development of the form

(1) H(z, z1) = SF (A-l(z, z1) + A"l1(z, z1) - An(z, z1) - An(Z1 Z)) n=1

where the A's are expressible in terms of the r.k.'s of W and W following the formulas from ?12. Since the kernel for W is K(z, z1), the kernel for W is immediately seen to be K(z, zi) +const. The constant is easy to determine and we get for the kernel K of W the expression K(z, z1) = K(zi, z) - 1/o where of is the area of D. All the terms of the development of H are then expressible by repeated integration in terms of the kernel K alone.

This development can serve to compute H when K is known. It will be especially useful when the domain is such that the subspaces 2t and W of e




form a positive minimal angle. This is equivalent to the fact that there exists a constant m >0 such that

mllfll < ?lf +-fill for any f Ez , fC Ez .

It is easy to see that the last condition is equivalent to the following property of harmonic, real-valued functions hEztC:

< c||kjj, for some constant c > 0,

where h is the conjugate harmonic function of h (that is, h+ih is analytic). This property can be proved for domains with a fairly smooth boundary (continuous tangent), with at most a finite number of angular points with positive angles (see K. Friedrichs, [1]). Consequently we can apply (1) to compute the kernel H for a rectangle, which computation would be especially useful in view of applications to rectangular elastic plates.

(2) We shall describe now a manner of applying our formula to calculate the kernels K.

Let C be a closed set in the plane (not necessarily bounded) of two- dimensional measure 0. Usually C will be composed of a finite number of arcs or closed curves. C decomposes the plane in a certain number of domains D1, D2, * - which together form an open set D. In D we consider the class of functions SD formed by all functions which in each of the domains Dn are analytic and regular. We call such functions locally analytic and the set C their singular set("3).

We shall suppose further that the functions of ED have a finite norm given by

(2) lIfl2 = ffI f(z) 2dxdy.

The class S1D possesses a r.k. KD which is simply expressible by the Berg- man s kernels KDn in the following way:

(3) KD(Z, z1) = KD.(Z, z1) if z and zl belong to D, KD(Z, Zi) = 0 if z and zi belong to two difterent Dn's.

There is no loss in generality if we suppose that C is bounded (we can use conformal mappings to reduce every case to this one). We shall denote by Do the domain Dn containing co.

Consider a decomposition of C in two closed sets

(4) C = C(1) + C(2)

To every C() corresponds the complementary open set D( and we have

(5) D-DM D= 2

(13) The author introduced and investigated this kind of function in his thesis in Paris [2 ].




Every function f(z) locally analytic in D is decomposable into two functions f(z) ==f(z) +f2(z) each of which is locally analytic in the corresponding domain D() (see N. Aronszajn [2]). Two such decompositions differ by a function with the singular set C(l) C(2).

The classes WD(i) are clearly linear closed subspaces of fD (when we restrict their functions to the set D).

The intersection of 1D(1) and WD(2) is equal to the class fD* where D* is the complementary set of C* = C(l) C(2).

The class fD* is reduced to 0 when the intersection C* is a finite or an enumerable set.

The equality

(6) SD = fDkl) 0 fD(2)

is not always true. For simplicity's sake we suppose now that C is not equal to 0 and is com-

posed of a finite number of rectifiable arcs or closed curves, not reduced to isolated points, and that the same is true of C(1) and C(2).

It can then be proved that the necessary and sufficient condition in order that (6) be true is that C* = C(l)* C(2)g 0.

Further, if C* = 0, then WYD [WD(1) EfD(2) ] is a one-dimensional space generated by the function w'(z) defined as follows: let h(z) hc(l),C{2) be the locally harmonic function in D, taking the value 1 on C(1) and 0 on C(2). De- note then by w(z) the function locally analytic in D (but not necessarily single-valued in the multiply-connected components Dn of D), whose real part is h(z). The derivative w'(z) = h'- ih is single-valued in D and belongs to 2D. h(z) is called the harmonic measure in D corresponding to C(1).

From what we said, it results that we shall be able to calculate the kernel KD by the formulas of ?12 in the following cases:

(1) When C* O and D?fD*#O we can apply the general formula (7), ?12, I, translating operators into kernels, in particular P, Po, Pi, and P2 into KD, KD*, KD(1, and KD(2). To calculate KID we have then to know KD*, KD(1), and KD(2).

(2) When C* O and SlD*=(O) we can apply formula (18) ?12, I. This case contains interesting particular cases; for instance in the case of

a simply-connected domain bounded by a polygonal line C. We can decom- pose C into one side C(1) and a polygonal line C(2) having one side less than C. This gives an inductive process to calculate the kernel KD of the open set complementary to C. By this process we obtain, at the same time, the two Bergman's kernels for the interior and exterior of C. This is especially interesting as these kernels will give the conformal mapping functions of the domain on a circle.

Another case in point is one when to a slit C(1) in the plane we add a rectilinear segment C(2), having only one point in common with C(). This




case may be of interest for the variational methods, as they were used by Loewner [1 ], for instance, in the problem of coefficients of schlicht functions.

(3) When C* = 0, then S=D* =O. We can apply formula (18), ?12, I, to compute the kernel of SD(1) 2D (2) if KD(1) and KD(2) are known. Then, if we know the function w'(z), we add the function (i/f w'ff 2)w'(z)w'(z1) to the obtained kernel and arrive at the kernel KD. The classes 2D(1) and 2S(2) have, in this case, a positive minimal angle for which a lower estimate can be obtained (using the constant m from the inequality m l fil 11f,+f211 as in ?12, I) by the use of methods developed by the author in [3].

If we consider the reduced classes S of functions of SD with single- valued integrals in every component domain Dfi of D, we shall obtain a r. k. KD expressible by a formula similar to (3) in terms of the reduced Bergman's kernels K' . For the reduced classes we have always f = 2(1)i D0(2) and in the case C* = 0, we shall calculate KD directly by formula (18).

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OKLAHOMA AGRICULTURAL AND MECHANICAL COLLEGE,

STILLWATER, OKLA.



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