On exponential representations of analytic functions in the upper half-plane with positive imaginary part

O N EXPONENTIAL REPRESENTATIONS OF ANALYTIC FUNCTIONS

IN THE UPPER HALF-PLANE W I T H POSITIVE IMAGINARY PART. ~*)

By

N. A r o n s z a j n and W. F. D o n o g h u e , Jr.

in Lawrence, Kansas, U.S.A.

Introd uction.

The work presented in this paper has its origin in certain theorems

which were needed by the authors for their forthcoming paper on "Finite

dimensional perturbations of spectral problems and approximation methods".

These theorems were closely related to results obtained by S. Verblunsky

[18, 19] and H. Delange [4] but could not be deduced f r o m t h o s e results.

Thus we were led to investigate in general the question of the exponential

representation of functions analytic in the upper half-plane with positive

imaginary part there, a class denoted in this paper by P. The results of

that investigation form the content of the paper, and the theorems which

we required for our future paper as well as the theorems of Verblunsky

and Delange occur as very special cases of those results.

Our work centers around the representation of functions tp (~') in l '

in the form

f l 1

where a~" O, ~3i is real and ~t is a positive measure for which (~.2q_ i)-~

is integrable, and also the exponential representation of such functions:

-J-O0

x

~ Q O

where ok is real and f(x) a Lebesgue measurable function for which

0 ~ f ( x ) ~ 1. Our main concern is the relation between the local behavior

of ~ at ~ . = ~ and the local behavior o f f ( x ) at x = f . The aspects of the

l .

583 (04). Paper written under contract with Office of Naval Research, Contract N-onr

321

322 N. ARONSZAJN and W. F. DONOGHUE, Jr.

local behavior of Ix which we investigate are essentially the following: the

presence of a point mass for ~t at ~. = ~, and the finiteness of the integrals

~-o ~+e

f i~.__$1 . and f i~.__~:1,' ~-~ ~+o

where ~ 0 and k is a positive number, the interval of integration being

open. The corresponding properties of f (x) when that function is regarded

as the density of a (positive) absolutely continuous measure also come into

consideration. In the investigation it turns out that the value of q~ (~') at

'.-" = $ is of importance. Our main theorems, Theorems A and B, describe

completely the relation between the local behavior of ~t and of f (x). Our

arguments are carried through in detail for the case $-~ oo and by an

elementary transformation are then brought over to any real and finite $.

Section 1 is a review of more or less well known facts about the

class P. This review has been made much more extensive than is necessary

for our work in view of our need for such information in our forthcoming

paper. The difficulty which we encountered in finding suitable references for

our statements, led us to suppose that such a collection of the essential

properties of functions in P would be useful to others as well as to our-

selves. The results which are assembled here are given without proof and

are usually found scattered throughout the literature in a variety of forms

and contexts. For some assertions we have been unable to give an adequate

reference, but these statements may be regarded as common knowledge

among the workers in the field and are easily deduced from the other

properties which we give.

We have not taken advantage in this review of the important paper

of Kryloff [11] which investigates the class of functions of bounded charac-

teristic in the upper half-plane, a class which of course is much more general

than P. It is regrettable that we have been unable to obtain the book of

Achiezer and Krein [1] which is reported to contain much that is of interest

about the class P.

Section 2 gives a description of the natural topology of the class P

~s well as some of the auxiliary theorems and lemmas used in the proof

of the main theorems.

O N E X P O N E N T I A L REPRESENTATIONS.. . 323

Section 3 is taken up by the proof of the main theorems A and B.

One of the auxiliary theorems of this section extends considerably a recent

theorem of I. S. Katz [8]. In Section 4 we give equalities and inequalities

between the moments and absolute moments of ~t and f ( x ) of integral

order k. These quantitative relations, joined to our main Theorem A, contain

as a very special case the theorem of S. Verblunsky [18, 19].

In Section 5 the results of the previous sections are brought over from

the point at infinity to any real and finite $. For the sake of illustration

two applications of these results are given: the first example consists of a

theorem of the type which provided the point of departure for our research,

while the second describes the change of the spectral measure of a singular

Sturm-Liouville problem when the boundary conditions are changed. In the

latter case our work depends on a theorem obtained by Gelfand and Levitan

[7] and sharpened by M. G. Krein [10], and is closely connected with the

results obtained by Aronszajn in [2].

In Section 6 we investigate certain properties of ~ such as the existence

of a point measure and the absolute continuity of ~ with respect to

Lebesgue measure, and obtain properties for f (x) which guarantee the desired

properties for B. The results are not complete; however, we obtain a con-

venient suffcient condition for the absolute continuity of ~ in an interval,

as well as a necessary and sufficient condition for ~ to be singular with

respect to Lebesgue measure.

1, Fundamental properties of functions in the class P.

As defined in the Introduction, the letter P will denote the class o f

all functions analytic in the upper half-plane and having a non-negative

imaginary part there. This class can be obtained in a natural way from the

class of all functions F (z) analytic in the unit circle for which the real part

is non-negative.

Let z = re ie' and suppose that h ( r , 9 ) = h (z) is harmonic and positive

in I z] < 1. It is well known that h(z) is a Poisson-Stieltjes integral:

2~

1 P 1 - - r 2

(1.1) h(z) = 2--W~ 1 + ~ - 2 r c o s ( , 9 - c o ) dr(co) o


where v is a positive measure on the interval 0 ~ o < 2 ~ t for which

2r~

, ( dv = 2~th(0)

0

and which is uniquely determined by h(z). The Fatou theorem asserts that

for any ro at which the density of v relative to Lebesgue measure exists

and is finite the limit in angle for z->-e it~ exists and is equal to that

density (see for example, [13]). I f h(z) is bounded then the measure v is

absolutely continuous with respect to Lebesgue measure and even has a

bounded density.

I f F(z) is analytic in the unit circle with the real part It(z) then

2~

1 f eU~ (1.2) F O) = - ~ - ~ ~ dv (co) + i t

0

w h e r e z - - I r a [ F ( 0 ) ] i s r e a l .

It is clear that any function of the form (1.2) is analytic in the circle and

has positive real part there. Many properties of such functions are given in

the literature; we shall state some of these presently in terms of the

equivalent class of functions P.

Let a = at + ia2 denote a point in the upper half-plane, (az > 0 ) , and

let r (z) be given by

a -- a z ( Z ) - -

1 - - Z

~ ( Z ) is a linear fractional transformation which carries the unit circle onto

the upper half-plane, and which maps z = 0 into ~" = a and z = 1 into the

point at infinity. Multiplying (1.2) by i and making the substitution ~'a(z)

in the integral, we obtain the class of all functions (p(~') analytic in the

upper half-plane and having a nonnegative imaginary part there: (2' s)

o.3) + (0 = + + ( [ i l--al 1

ix_al21 a O.) d [

2. All integrals occuring in this paper without explicit limits of integration are to be taken over the entire real axis.

3. The passage from (l.2) to (1.3) is carried out in detail in [12] for the case in which a = i .

ON EXPONENTIAL REPRESENTATIONS...

where

(1.3') a ~ 0 , [3~ real, i TM d~(~,) _ j 1 2 + 1 < o 0 .

1 The coefficient a is positive and equals 2u v [0] , where v [0] is the mass

which v concentrates at ca----O, while the measure ~t is defined by

at~ (x) = ! ! t - a 42 av (co (1) ) ,

co(l) being the inverse substitution which carries the real axis onto izi = 1.

Thus I i - - a [-2 is integrable with respect to the measure ~. The integral

in (3) is a Lebesgue-Stieltjes integral, although the separate terms (~.--~)- '

and ( 1 - a l ) J t - - a 1-2 may not be l t-integrable. It is easy to verify that the

constant i3~ is the real part of ~p(f) at the point f = a .

We say that (1.3) is the canonical representation of q~(O in P relative

to the point a (briefly: canonical representation rel. a). It should be

emphasized that the coefficient a and the measure ~ are quite independent

of the choice of a while 13a depends on a in an obvious way. It is usual

to take the canonical representation relative to a = i obtaining

f l 1 ~" 1 (1.4) q) (~') = ~ + ~i + ~ l - - ~ 12+i d~t (1) .

This establishes a one-to-one correspondance between the set of all functions

in P and the set of all triples [ a , [ 3 i ,~ t ] where a~>0, ~# is real and ~ a

positive measure for which (~. '+ 1) -~ is integrable.

Let a, be a point on the real axis and let ~t be the measure

diminished by the point mass #[a j ] at i = a , . In the spedal case that

{ ) . - - a , ]-* is integrable with respect to ~ in a neighborhood of 1 = a, we

may write

(1.5) q ~ ( ~ ' ) = ~ ( ~ - - a l ) - + ' ' 3 a , + ~[a , ] - J - f [ 1 1 i ~ , - : Y~-c - ~--,~, dfi(~)

a ~ ( t ) F ( 1. 5 J)

�9 r V ~ + 11 t - ~, I

obtaining a canonical representation rel. a~.

�9 e integral f l ~ i ( ~ ' + ~)-- 'd~(1) is ~nite rise to a canonical representation rel. ~ :

It is also easy to see that if

the representation ( 1 . 4 ) g i v e s


d~ (~) ~d~t (~) (1.6) q~(~')=(x:. + ~ + , t ),__: , ~ : ~ i - - , / t k2_{_1 real,

0.6') f < The representations (1.5) and (1.6) were obtained under the special hypotheses

(1.5') and (1.6') respectively; however, if the integrals are to be Lebesgue-

Stieltjes integrals it is evident that these hypotheses are both necessary and

sufficient for the existence o f such representations.

We shall often make use of the simple identity

1 ~ ~ ' + l 1

~--~ ;~2+1 ~--~ ~ 2 + 1 �9

For further study of functions in P we shall require the concept o f

convergence in an angle. For any real ~ the expression "~"-~ ~ in an angle"

_ _ r converges to ~ remaining will mean that for some ~ where 0 < ~ < 2 ' "

inside the angle ~ < arg (~" - - ~) < u - - ~. The point r will be said to converge

to infinity in an angle if it converges to infinity within the sector

~ < a r g . ~ < u - - ~ . In either case the convergence will take place "in any

angle" if F. may be taken arbitrarily small. W e make use of this concept

to extend the definition of functions q~ (~') in P to the real axis (such functions

are initially defined only in the upper half-plane). For any real $ at which

the limit as ~" approaches $ in any angle exists we set q~ (~:) equal to that

l imit ; the function will be left undefined for other real points $. It will

presently appear that the extended function is defined almost everywhere on

the real axis.

The r~easure I~ corresponds to a monotone increasing function on the

real axis which we normalize as fol lows:

(0) = 0 , ~(~) = ~ (~+) + ~ ( ~ - ) 2

[A] will be the measure of the set A, in particular F [$] will be the mass

which F concentrates at the point ~:. (This will not be confused with I~($)

the value of ~t(~.) at ~. = .~.)

For the more detailed study o f q~(,~) we introduce the following

functions ($ is a fixed real number ) :

ON EXPONENTIAL REPRESENTATIONS... 327

' 2~

Note that ~ ( i ) ~ 0 and that the left and right derivatives of bt(~.)

at t = 6 : exist and are finite if and only if lira x~(~.) and lira tg~(~,) ~. +0 ~,-->.o

both exist and *tre finite and ~ [~] = O.

If q0(~') in P is given by one of the canonical representations (1.3)

(1.4), (1.5) or (1.6) and 6: is a real number, by S~0(6:, h) we shall under-

stand the number given by the same representation of q0 computed for ~ '= 6:,

the integration being taken only over the complement of the interval

~ - - h < ~ , < 6 : + h . S~0(6: , O) will be lim S~(6:, h) (if the limit exists) and /~-+o

will therefore correspond to a singular integral.

One can establish the following inequalities without difficulty:

(1.7) lim inf ~t0~ (~) ~ lim inf llm [q0 (6: + D2)] - ~t ~ ] } ~--~o '~-->-o

<___ lira sup {Ira [q0 (6: + i ~ ) ] - - ~t [6:] } ~_ lira supzt{~ (~),

(1.7') lira sup {Im [tp ($ -q- i~)] - ~ [$ ] } ~ lim sup 0g (~) . ~--~o ~l -~--~o

If A[ (~/) = Re [q~ (~: + i~)] - - S~ (~ , r~), then we have:

(1.8) lira infS~ (6:, *2) ~ lira infRe [q~ ($ + i~/)] ~ lira sup Re [q~ (6: + it/)] ~-~o ~-~-o ~-~o

lira sup S~ (6:, ~l), ~-~o

(1.8') --2 [lim sup x~ (~) -- lim infx~ (~)] ~ lira inf Ag (~) ~ lim sup Ag (~l) ~->-0 ~->-0 ~-->-o W-~O

<_~ 2 [lim sup x~ (~/) -- lim infx~ (~)]. o) w-~0 ~--~0

4 These properties are immediate consequences of the following relations which are easy to obtain. Without loss of generality one may suppose ~ =0 and

1

q~ (~') = ~_~.

Now - - t t/~

2~'9~ (1) f 4~'2 0~ (~)') d~' ~t[O] 9o(O+~[~ < ~ [ ~ ( r + ! + - - - - ,

= 1 + ~2 ~ .. (V + 1) ~ ~l o


We now list certain important properties of functions q0(~) it~ P. (s)

I. cI = lim q~ (~') / ~ when ~ -->- oo in any angle [20].

II. For any real ~ l im(~- -~)q~(~)=~l [~] , when ~->-~ in any angle.

III. If q~(~) admits a canonical representation rel. a, where at is real

and if [~ [a~] = 0 then lim q~ (r) exists and equals I~a~ when :~-->-a~ in any

angle. If q~ (~) admits a canonical representation rel. c,~ and if r then

limq~(~') exists and equals ~ when ~"->-c~ in any angle. b

IV. ~t (b) -- ~t (a) = n.-~01im f ~ r Im [~p (~ + i~)] d, # , [5, 1 3]. g 6

V. If Im [(p (~)] ~ M throughout the upper half-plane, then c t= 0

and ~t is absolutely continuous with respect to Lebesgue measure, the density d~t dl being bounded by M ~ .

VI. If lira (p (~+i~) exists, then, even when the limit is infinite ~q.-~0

lira (p (~') exists when ~ - ~ in any angle and q~(,#)= lira q~(O is defined [13].

VII. If lim • (l) -- xr (O) exists and is not 0 then lira Re[q0(~+i)/)] ~.-->-0 ~.-~-o

exists and equals + ~ when x~(O)>O, and equals --oo when x~(O)<O.

VIII. For almost all ~, S,(~,O) exists and is finite and lim Re[q~(~-ki~/)] ~-->0

exists and equals S r whenever the latter exists [3, 14, 15, 17].

IX. For all ~, lim Im [q~(~+i~/)] exists and equals lira atg~(~.) ~--~0 l--)-0

while

and

1/~ Re [~ (i01 = / 2S~ (0 , ~1) d l

0

1

7/2 �9 1__12 Ao(~) = 2Xo (~) l+~l 2 po(1)- __.~ --(I+~'2~-'--~ 2~.Xo (~i) d~,

o

~" 312 + i 2X0 (7/l) dl J 1 (I 2 + Iy ]

5. The references will frequently establish~the property of interest only for

functions F(z) in the unit circle, but the transition to the half-plane is immediate.

For some assertions we have been unable to find an adequate treatment in the literature ;

the proofs are nevertheless omitted since the results are not hard to obtain.


if the latter limit exists, while lim s u p O [ ( q ) ~ + o0 is equivalent to ~-->-0

lim sup Im [q~ (~ + iT)] = + oo. *}--~o

d~t X. For all ~ for which ~d~-- (~:) and S ~ ( r O) exist and are finite,

d~t ~9 ($) = lira q0 (~ + i~/) = S~ (~, O) + in - ~ (~). This holds almost everywhere.

~-~0 XI. The function ~ [ q0 (• + iT)[ is uniformly bounded in any rectangle

0 < ~ / < h , a~_$gb. XII. Let a and b be real, a < b ; then ~t (b) :: ~t (a) if and only if

r can be continued analytically across the interval a<~r<b in such a

way that the extended function satisfies c p ( ~ ) = ~ ( f ) . In this case the

extended q~(~') will be regular, real, and increasing on the interval and will

be given there by S~ (~", 0 ) , the singular integral being in fact a Lebesgue-

Stieltjes integral.

XIII. If ~(~) is absolutely continuous on a < ~ . 1 then S~( r O) (which

exists almost everywhere) is in LP(a', b') for any a' and b' such that

a<a'<b'<b. This is an immediate consequence of a theorem of M. Riesz [16].

If f(x) is in LP and Hf(y) is given by the singular integral ; f(x)dx �9 x - y

then there exists C > 0 for which

j"l f(y)l,ay<= c f l/(x)lpax. XIV. For any p < l and any finite interval a<~<b, S~(~,O) is in

LP(a,b). This is essentially contained in a theorem of Kolmogoroff [9]

which asserts that there exists a C>O such that for f(x) in L ~ and p < 1 b b

C f [Hf(Y)]t'dY~ 1---~f [f(x)!dx"

The next property is of quite recent origin and is due to I.S. Katz [8].

XV. A necessary and sufficient condition that ct = 0 and that

[~,]*tl < ~ for some * in the interval 0 < * < 2 is the finiteness of

the integral --./" lm [q~ (iT)] d~.

1

3~o N. ARONSZAjN and W. F. DONOGHUE, Jr.

It is convenient at this point to give the triple [c~, f3~, bt] for certain

particular functions in P ; we shall write d~ to denote Lebesgue measure

while X+ (X) and Z-(~) will be the characteristic functions of the positive

and negative half axes respectively. The function Log~ is the principal

determination of the logarithm and is a function single-valued and analytic

in the plane slit along the negative real axis which is real on the positive

half-axis.

(~.ga) qo(O = a + ib, a real b ~ O: [0 , a , b / = a q

(1.9b) ( p ( ~ ) = ct~ r + { 3 , ct>_O, ~ real: [ ( z , ~ , O ] .

(1.9c) ho(~') = - - 1 / ~ : [ 0 , 0 , unit mass at ~.----0].

For 0 < y < 1,

(1.9d)

(1.9e)

0 . 9 0 (,.gg)

(1.9h)

(1.9i)

q~(.~)=rv-- '--exp(yLog~):[o cos(~'a/2) 1 ,ki~sin(yz) Z_(~)d~ ]

sin (r~) X-(~) d~] q, ( ~ ) = - - 1/'~,t - - --exp (--y Log ~') : {0, --cos (~'~r/2),

(p (~') = tan ~" : [0 , O, unit mass at poles of tangent].

(O = log ~ - Log ~ : [o, o , z - (x) ax]. q~(~) = --log ~" ~ --Log ~" + 2ai : [0 , O, (1 + ; C (~))d~].

(9 (~') = log (-- t l~) ---- Log (-- t[~') : [0 , O, Z + (it) diq.

two functions are most conveniently taken in the canonical The next

representation tel. oo; a and b are real with a < b .

(l.9j) (?(.~)---- ..r_ a : [0 , ~300= 1, point mass ( b - - a ) at ~,= a],

O.9k) q~ (O = log T ' - T - Log ~ : [ o , ~| = o , X~,~(1) a~]

Xa. b (X) ----- characteristic function of a < ~, < b.

where

While the class P is obviously not a linear space, it is closed under

addition. More importandy, it is also closed under composition: if ~ ( r )

and q~(~') are in P so also is xp(q~(O). Since --1/~" is in P it follows

that --1/T(~') is in P for any q0 in P, also q0(--1/~') is in P. Moreover,

taking the principal determination of the logarithm, Log q0(~') is also a

function in P. Since Log ~" has a bounded imaginary part in the half-plane

it follows that Logq)(~) also does, and therefore that the representation

(1.3) for Log CP involves a coefficient ct-----0 and a measure ~t(~) which is


not only absolutely continuous but has a bounded density. Thus

dbt (~.) = f 0.) d~.

where fQ.) is bounded and measurable. Since f ( ~ ) is almost everywhere

equal to lira 1_~ Im [Log q0 (~.+i~)] it follows that 0 ~ f (Z) ~ 1. Accordingly ~--~.0 ~

we may write

( 1 . 1 0 ) .(O=exp[o,+ f [. I''""I' where oi is real and f(x) is measurable, 0 ~ f(x)< i . Clearly

o, = Re [Log q~ ( i ) ] .

We call (1.10) the exponential representation of q~ in P. Every

function in' P except the one which is identically zero has such a represen-

tation and there exists in consequence a one-to-one correspondence between

such functions in P and the class of all pairs [ o i , f ( x ) ] . It is clear that

the functions f(x) appearing in (1.10) are determined only up to a set of

Lebesgue measure zero; we ignore this ambiguity in the sequel and shall

speak of "the" f(x) associated with q0(~').

If q~(~') is real and regular on the interval a<r~<b it follows from

the definition of f(x) that f(x) assumes only the values 0 or I in a < x < b.

Thus f(x) is 1 where q~(~') is negative and f(x) vanishes where q~(~') is

positive. A function q0 ( f ) in P which is meromorphic in the entire plane

and real on the real axis has real and simple zeros and poles and cor-

responds to an f (x) which is the characteristic function of the set upon

which q~(~') is negative; the measure I~ is a distribution of mass at the

poles of tp(~'), in fact, if ~,0 is such a pole, ~t [t0] = - - r e s id u e of q0 at ~o.

It is now clear that functions f (x) which are the sums of finitely many

characteristic functions of disjoint intervals correspond to rational functions

in P and that the rational functions in that class which are real on the

real axis correspond to such functions f (x). The endpoints of the intervals

in question are the zeros and poles of the rational q0.

The following property is due to S. Verblunsky [18, 19] (see also

H. Delange [4]).

f aN(q XVI. q0(~ is representable in the form 1 + Z--~

finite total mass if and only if

with t~ of


f f (x) ax ,p (r : exp x----2

with f ( x ) in L t and 0 ~ f ( x ) ~ l ; in this case

+oo +ao

2. Topology of P; Factorization of functions in P. If the class P is provided with the topology of uniform convergence

on compact subsets of the upper half-plane then with the obvious choice

of a uniform structure P appears as a complete metric space which is

separable and locally compact. If % ( 0 is a sequence in P converging to

% (.L'), we have the convergence of % (i) to % (i) in particular, and making

use of the representation (1.4) we infer the convergence of the sequence

/~/~,) to [3 (~ as well as the convergence of

It follows that the functions ~, (~.) are uniformly bounded on intervals of

the form l~l <=N and by Helly's theorem, the sequence ~t,(~.) converges

to ~ (~-) for all ), except perhaps at the points of discontinuity of ~ (~.).

Thus the convergence in P implies the convergence of the sequences 13( '0,

[ f a ' l % + . 7 ;Lzq-1 ] and I~,0.). Conversely, the convergence of these sequences

implies convergence in P since this immediately Implies the pointwise

convergence of the sequence

,r,. ( 0 = H(~ ", +.-[~. + j ~ J l I*

while from the boundedness of ~3-t-1 it is easy to infer the uniform

boundedness of the sequence q% (O on compacts, and therefore the uniform

convergence on compacts.

From the nature of the convergence of the ~n(~.) above it is not

difficult to infer that

f _ _ d ~ ( ~ ) dgo (x) ( lira inf ~ ~z ~.2 + 1 - - n-.-)-oo ,,-" + 1

and therefore that lira sup an ~ So. The sequence a,, will converge to %

ON EXPONENTIAL REPRESENTATIONS... 3 3 3

T

when q~.(~') converges to q%(~') if and only if the integrals ~.2+ 1

~ T converge for z--~ oo uniformly in the parameter n.

We should remark also that if q:,(~') converges in P to q~0 ( ( ) in

such a way that for some l ~ l the numbers (~t!--d~t,,0,)~- are uniformly - - , ! ~ . +1

/ ' l~i ~ bounded by M, then {x, converges to a0 and . ! ~2q_ 1 d~t~ (~) converges

14 [~'[~ d i ~ I~'lt d~t0 ~ M. We have also to J ~ ~t00.) forall o~h<l , while j ~2+1

f L~I' i f and only if i ~' iXl' d~. (~) converge for �9 + ~o ~niformly in n. d ~.2+I

I f a convergent sequence q~(~') consists of functions whichcan be

written in a canonical representation rel. 0% then the limit q~0 (r) can also

so written if (but not only if) the integrals .J--,,T(~]I dS"0") are uniformly be

The corresponding ct,, f~) , and ~[2~1d~,, converge to %, bounded.

T"

13~ ) , and j dbt0 respectively if and only if d~t,, converge

for z-> oo uniformly in n. Similar statements can be made when the

functions admit a canonical representation tel. a for any finite real a.

It is easy to see that the convergence of a sequence in P which does

not converge to 0 is equivalent to the convergence of the logarithms, which

in turn is equivalent to the convergence of the numbers o(i") and of the

measures f,,(x)dx. This second condition means that the functions t

F,(t) = t fn(x) dx f f

0

are uniformly converging on every finite interval.

We have seen that for ip(~) in P the functions tp(--1/~') and cp(~'+a),

a being real, are also in P. It is not difficult to verify that P also contains

the "reflection" ~ (~) of ep (~') defined by ~ (~') = -- W (--~)- It is important

to obtain the representations (1.4) and (1.10) for these functions.


If (p (~)

shows that

is given by (1.4) and (1.10), a straightforward computation

x_.~ ~2+1 a~(x)

= exp{o,+ j ' [ 1 x x--* x2+l ] [t - f (-x)]dx}

where ~ is the reflection of I* through the origin, that is, ~(JL):--!~(--~.).

Again, if , (~') is subjected to a translation to the right or left by a

real number a giving rise to q:*(~')= q0(~+a) the corresponding represen-

tations are easily found:

~" (x) = ~ (~ + a) - ~ (a), /* (x) -- f (x + a).

The coefficient ct is the same, while

[3* ---- Re [~0 (i + a)] and o* ~ Re [log q~ (i + a)] .

A slightly more complicated change of variables enables us to find

the representation for q~0 (~) = q~ (--1/~') :

(2.2) q~o (~') ----- I* [0] ~ + l~i + - - .~ - + . ~._~.. ) 2 + 1

where, absorbing the mass (L at the origin into ~0 we obtain finally the

measure given by

(J0 = F t-2d~t(0 + ct /2 f o r J~>O, P.0 /

+oo

po(X) = ~ t" t -2d~t( t)--ct /2 f o r ~ < 0 .

- - l l ~

The corresponding fo(x) is given by f e (x )~- f ( - - l /X) , and we have

(2.2') ~po(~') ---~-~ exp o i + x--~ x 2 + 1 f ( - -1/x)dx .

The exponential representations given by (1.10) are conveniendy

extended to provide representations for functions which are not in P but

which are closely related to functions in P. Two particularly useful ones

follow.

If ~0(~) is given by (1.10) we may write q~(~')/~ in the form


(2.3) 'V (.C')/( = exp [o, + j "

Similarly,

(2.4) -~'qo(O -- exp [o,+ f

x - ~ x 1 xZ+ 1 ( f (x) -- Z- (x)) dx .

x I x~+l ( / ( x ) - z + ( x ) ) d x .

Suppose {p'(~') and q0"(~') are two functions in P ; in general their

product is not in P, however, it is easy to see that the exponential rep-

resentation of the product exists anyway and involves the function

f (x) = f" (x) + f " (x).

However, we have the obvious

L e m m a 1. q~(C')=(p'(~)cp"(~') is in P i f a n d o n l y i f

f" (x) + f" (x) ~ 1 .

We introduce two special classes of functions in P : P ' will denote

the class of all functions q)'(~') in P for which the corresponding f ' ( x )

vanishes on the right half-axis, while P* represents the dass of functions

in P for which f " (x) vanishes on the left half-axis.

L e m m a 2.

f [ ' ,p'(~) = ,~'~ + [Y~+ -~_~. ~2+ 1 dffO.)

b e l o n g s to P' i f a n d o n l y i f t h e o p e n l e f t h a l f - a x i s is a o

f d~'(~) s u p p o r t f o r ~', t h e i n t e g r a l , ~.(~.2+1 ) is f i n i t e a n d

0

(2.5) ,p'(O) = [3'i + j.(~2+1 ) > O.

In t h i s c a s e

(a) lira q~'(F)/~ = , [ ; ~-~+c~

(c)

0

(b) - ' > 0 i f a n d o n l y i f f x [ f ' ( x ) - z - ( x ) ] d x < ~ ; x 2 + 1

q)'(~') is b o u n d e d f o r ~ > 0 i f a n d o n l y i f r [ = O a n d 0

12+ 1 - - 0 0


in t h i s c a s e ~9'(~') a d m i t s a c a n o n i c a l

rel. oo w i t h {3'~ = lira q/(8) > 0 ; ~-->.+oo

(d) ~9'(~) i s b o u n d e d f o r ~ > 0 0

f Ixlf'(x)ax x ~ + t < c~;

(e) f o r a l l ~ > 0 a n d a n y p o s i t i v e ~

o ~ 0.'r < ~' ( 0 <= w' (~:) + ~ la' +

r e p r e s e n t a t i o n

1 convergence] of ~.-~ positive values.) This

(a) we note that

i f a n d o n l y i f

0

ap,' 0.) f ~,(~-~) ] �9

Proof. If p' satisfies the conditions of the lemma the corresponding

f zr d~'(~) t~'(Ot$ = -' + ~',/$ + ~(~-0 z~+/- ;

dp'(~O has finite total mass and the integrand -~(~_ff) the measure ~.2+1

converges pointwise to 0 as ~ approaches + ~ while it is bounded by the

integrable 1 + 1/i~,i. Since cp'(~)/~ may be written as

(9' (~) is regular on the right half-axis and increasing there, moreover q/(0)

f dl.(O0 exists and equals p ' i + ),(~.2+1 ) and is therefore nonnegative. Thus

q3' (~) is positive on the right half-axis from which it follows that f ' (x)=O

for x >0 , i.e. q3' (~') is in P'. Conversely, if q3' (~') is in P' the function

is positive, regular and increasing on the right half-axis, whence the support

of I~' is the closed left half-axis. As ~" approaches 0 through positive values

the limit of ~p'(~') exists and is nonnegative; this limit is also equal to

lira [3'~+ ~._~ ~2+1 1 lYe+ ~.(~.2+i )

[. d~' (~): and therefore the finiteness of the integral j ~. (~2+ 1) immediately follows,

as well as bt [0] = 0. (Here we have made use of the monotonicity of the

~ - + 1 to ~t(~3q-1) as ~ approaches 0 through

proves the main assertion of the lemma. To prove


it follows that

0

[ f-+, exp o ~ + -x-~ ~- (f" (X) - - Z-- (X)) dx ] x * + 1 !

0

lira rp'(~)/~ = exp (oi) exp -- x 2 + 1

and ~ ( = 0 if and only if the integral occuring in the exponential is + ~ , 0

i.e. a' > 0 if and only if ; x . [ / ' (x) -- ;(- (x)] dx is finite. This proves (b). �9 x ~ + 1

For the proof of (c) we suppose first that q/(~') in P ' is bounded on the

right half-axis; it is obvious that a ' = 0 and since the function is increasing

for ~ '>0 , 0

sup q/(~) = lim cp' (~) --- ~'i + lira f ~'~ + 1 d~t' O0 ~ > 0 ~ - ~ + o o ~--~-+o,~ --~ ~. - - ~ ~.2 + 1

0

mOO

is finite. We may then write

-~d~' O0 ) ~ 2 + I

qr (0 = Y| +

0

~.--~

and

~'~ = lira q0'(~)>O,

1 since ~ _ ~ converges monotonically to 0 for ~.<0. Conversely, a function

~p' (~') in P' which is of this form is clearly bounded on the positive half-

axis by ~'| We could also compute this bound by making use of the

exponential representation of (p'(~') and taking the limit as ~ approaches

-boo, obtaining o

[Y| =exp(o~)exp ~ x ~ + 1 "

The integral in. the exponential is then finite if and only if q/(~') is

bounded on the right half-axis. Thus (d ) i s proved, while the inequality


(e) follows from 0

,p' ( 0 - ~' ( 0 = ~' ( ~ - 0 + j ~ - ~ dr,' (~) ( ~ - 0 (~-0

~ a O

0

- ~ ( ~ _ ~ ) ] - ~ r

Lemma 3.

,~ (r = eL"'.. + fs"i + ~ . _ f ~.2+ 1 . arfa" (~.)

b e l o n g s to P" i f and o n l y i f ct"=O,

ax i s is a s u p p o r t fot ~t", the i n t e g r a l ~2q-1 0

0f l

~d~" (~) f~.| = fj,.,_ [ ~.2+ 1 J

t h e c l o s e d r i g h t h a l f -

is f i n i t e and

is n o n n e g a t i v e . In t h i s c a s e

(a) lira r ~"~ ; ~->.-oo

0 0

(b) ~ " ~ > 0 i f and o n l y i f . i - - 0

(c) f o r r < o ,

xf"(x) dx is f i n i t e ; x2-b 1

~t"(~) ~q / , ( . . . )~ I~"(~) [ / d~"O.) ] o ~ 82 + - ~ _ ; - _ - l - T r / + ~; + z

7

w h e r e T is any p o s i t i v e n u m b e r .

Proof. If ~t" satisfies the conditions of the lemma and ct"= 0 with

[5~___ 0, then the corresponding q/(~') is regular, positive and increasing

on the left half-axis, whence f" (x)-----0 for x < 0 and q/(~') is in P*.

Conversely, if h0" (O is in that class, it is regular, positive and increasing

on the left half-axis and the infimum there is found by taking the limit as

approaches --oo of et"~ + ~3"i + . f ~ + 1 dbt" (k) a limit which ~- -~ F + I ' o


can be finite only if ~" = 0, and which in that case will equal

i Since this is finite and positive it follows that ~"i + ~.~ + 1

0

ep" (~') admits a canonical representation rel. co. Assertion (a) and the fact

that J3~ ~ O are now evident. We can also compute [3'-' by finding

lira q0" (~ )= lim e x p ( o i ) exp x - - ~ x 2+1

~ x f ~ (x) = exp (o"i) exp j " dx

�9 __ X 2 + l o

and therefore ~ > 0 if and only if the integral in the exponential is finite.

An elementary computation provides the inequality (c).

It is obvious that any function q~ (~') in P has a factorization which

is uniquely determined up to a constant multiplier into a factor q0'(~) in

P' and qo"(~) in P" where qo'(~') corresponds to f ' ( x ) = f ( x ) z - ( x ) ,

while q9" (~') corresponds to f " (~) = f (x) Z + (x).

Theo rem I. I f

( 0 = qr (~') qD" ( 0

w h e r e qg'(~') i s i n P ' a n d q~"(~) i s i n P", t h e n

/ (x) = f ' (x) + 1" (x), a = ~' ~

a n d

dr (x) = ~' (~) dr" (x) + q~" (x) dr,' (x).

Proof. f = f ' + f " is obvious. When ~"-->-o~ in angle we have

c t = lim cp (~')/~" = lira q/(r)/~" lim c p " ( ~ ) = ct'[5~.

(See I and III, Section 1, and Lemma 3.) Moreover,

r` [0] = lira -- ~'q~ (~') = lim [--~rtp" (~')] lira ep' (~') = r`" [0] q~' (0) ~;r ~--~o ~;-->-o

in accordance with the given formula. Next let 0 < a < b; we will show that b

(b) - r (a) = . j ' q~'(~) at" (~). r` tg


From IV, Section 1, we have b

1 j " ~ ( b ) - ~ ( a ) = lira ~ - [v'(~ +i~)u"(~-t- in) + v"(~+i~)u'(~+in)]df,

a

where

cO'(O-= u'(O + iv'(O and C ( O = u " ( 0 + iv " ( 0 �9

We may write

2)' v' (~ + i~) u" (~ + iO = (~ + i,2) ~u" (~ + i O . n

As 7~ converges to O the ratio v'(~ + i~) converges to the continuous 7~

0v' function - ~ uniformly in a <__ ~ <_ b since v'(~) is harmonic in a region

containing that closed set and vanishes on the real axis. From II we have

~" [$] = lira ( - ' /7) q0" (~ + i77) = lira (--iT) (u" + iv ') , �9 Q--~O ~--~.0

whence

lira ~u" (~ + i~) = O . ~-~0

Since by XI, Section 1, the functions ~u" (~ 4-i~) are uniformly bounded

in a ~,~<=b, 0 < 7 < 1, it follows that the functions

~' ~ + i~) u" (~ + i~) = g~ (~)

are uniformly bounded in a ~ b and converge pointwise to O. The

bounded convergence theorem then provides that

b

lira 1 j v ' ($+iT) u " ( 3 + i T ) d ~ 0 7-.>.0 ~t ,

a

We have found that b

~ , ( b ) - - ~ ( a ) = l i m 1 _ (v" (~+i~?)u ' (~+iT~)d~ . �9 ~ - - > . o ~

It is clear that u ' (~ + ir~) converges uniformly on the interval to

since

hand,

fp, (r) is regular in a region containing that interval. On the other

I v " (~ + iT) is the density associated with the function 7g


, n (O --- ~: (~ + in), i.e. a,% (x) = x__ : (~ + in) ax. ~t

*~(~') is in fact in P and ~t~l is absolutely continuous with respect to

Lebesgue measure. As ~2 approaches O, q,~ ( 0 converges in P to tp" (~'),

hence the measures ~t~ converge to F" on the interval a ~ ~. ~ b

Accordingly, b b

pt (b) -- ix (a) = lim u' (~ + i~) d~t~ (~:) = o 2' (~.) dgt* (~), ~-->-0 a

as desired. Precisely the same argument goes through for intervals a < b< O,

thereby establishing

dr C ~) = 'p' (~) d~" (~) + , : (Z) dN' (Z).

Theorem II (Comparison Theorem).

(a) L e t q91(~) a n d q~2(~') be t w o f u n c t i o n s in P s u c h t h a t

on t h e i n t e r v a l a<r~<b t h e d i f f e r e n c e q0t(~')--q02(r) is

r e g u l a r a n d p o s i t i v e ; t h e n fl(x)~_fz(x) a l m o s t e v e r y w h e r e

in a<x<b. (b) L e t q~l(~') a n d qg,(~') be t w o f u n c t i o n s in P s u c h t h a t

o n t h e i n t e r v a l a<~<b t h e d i f f e r e n c e Logq)a(~')--Logq~2(~')

is r e g u l a r a n d p o s i t i v e ; t h e n d~2"Cdp.t on a<~.<b, i.e.

bt2[A]~bta[A] f o r a n y B o r e l s u b s e t A o f t h a t i n t e r v a l .

Proof. (a) For almost all ~ in the interval a<~<b the four functions

qh (~), q~2(~) , Logq91(~), and Logqoz(~) are defined, and by hypothesis

Im [qo~ (~)] = Im [q02 (r while Re [q01 (~)] ~ Re [~0z (,#)]. It follows that

the phase angle of q~l (~) is not greater than that of (P2 (~:) and therefore

that f l ( x )~ f2(x ) almost everywhere in the interval.

(b) We note that the logarithm of the ratio (~t (r)/(P2 (~) is regular

and positive in the interval; therefore the ratio itself is regular and positive

in that interval, in fact, > 1 there. Thus q~l (~') = H (~') q~2 (~') with H (~')

regular in a region containing any subinterval of a<r~<b. Write

n {~) ---- R (~) + iI (~) ;

the functions R(~') and I(~) are harmonic in such a region, 1(~') vanishes

on the real axis and R (~') > 1 there. Now, vt (r) = R (~') v2 (~) + I(~') uz (~')

and by the same argument used in the proof of the previous theorem the


product

! ( 0 u~(O = l (~ + i,~) ,~u2 ($+i~)

converges to 0 as ~-->-0 for all $ in the interval and boundedly for any

closed subinterval. Moreover, on any such subinterval R(~ + i~) converges

uniformly to a limit > 1. Accordingly, if a < a'< b'< b, b~ b~

~q ">'0 ~ J 7/'->'0 ~ d ad a~

b *

> lira 1----( v, (~ + ir~) d~; --- ~tz (b') -- ~t2 (a ' ) . ar

3. Main Theorems .

We introduce the following classes of measures: for any pair of non-

negative numbers h and k, ~ (h , k) will denote the class of all positive

or negative measures ~t on the real Laxis for which

0 + ~

(3.1) ~ar~ O

Inclusion relations of the kind 9X (h + 1 , k) C ~ (h , k) obviously hold.

The class ~ + ( k ) will consist of all measures in ~ ( 0 , k) which vanish

on the left half axis, while 9X-(h) is composed of measures in 9X (h , O)

vanishing on the right half axis. We make a similar classification of functions :

@(h , k) consists of all measurable functions f ( x ) for which I f (x) I ~ 1

and such that

0 + ~ / ~ I~ I h ~

x-r-~--~lf(x)[dx + f x-r-~TlfCx)!dx < oo. (3.z) , J

~ oD o

We define {~+(k) and @-(h) similarly. Note that for h and k smaller

than I we have @- (h)---- (~- (0) and (~+ (k) == ~+ (0) . Finally we define

~ - ( * ) as the class of all functions f '(x) which together with If' (x) l - ) ~ - ( x )

belong to @ - ( O ) - - @ - ( I ) , (~ and @+(~ as all functions f ' (x ) which

together with ! f " (x) I - - Z + (x) belong to ~+ (0 ) - - ~+ (1).

6. ,4 -- B denotes the set-theoretical difference of A and B.


Theorem A. L e t q~(~') be a f u n c t i o n in P. In o u r s t a n d a r d

n o t a t i o n p , / e tc . a r e t h e c o r r e s p o n d i n g e l e m e n t s o f t h e

t w o r e p r e s e n t a t i o n s o f q~. F u r t h e r , l e t h a n d k be ~ 1 ; t h e n

(a) / (x) -- X- (x) b e l o n g s to @ ( h , k ) i f a n d o n l y i f ct>O

a n d ~ is in ~ ( h - - t , k - - I ) .

(b) f(x) b e l o n g s t o ~6(h ,k) i f a n d o n l y i f ct-~O, ~t is in

~J~(h,k) a n d ~ |

(c) f ( x ) - - I b e l o n g s to ~(h , k ) i f a n d o n l y i f el=O, V is

in ~J~(h,k) a n d ~ < O.

(d) f (x)-- X+ (x) b e l o n g s t o (~(h,k) i f a n d o n l y i f e l=O,

~t is in ~ 0 ~ ( h + l , k + l ) a n d ~oo-~0.

For the proof of Theorem A we need a series of lemmas and auxiliary

theorems. We start with a lemma which settles the simplest case of

Theorem A, (b).

L e m m a 4. f(x) b e l o n g s to @(l , 1) i f a n d o n l y i f , x=O,

Vt is in ~IJ~(1, 1) a n d ~ > 0 .

Proof. In view of Lemma 2, Section 2, parts (c) and (d) and

Lemma 3 of that section, particularly part (b), the lemma holds for functions

in P' or P". If f(x) is in @ ( I , 1) we set f ' ( x )= f ( x )X- ( x ) and

f " ( x ) = f ( x ) X +(x) to obtain functions q0'(~) and t~"(~') in P' and P"

respectively, the product of which, except for a positive constant multiplier,

will equal q~ (~). We may suppose that the multiplier is 1. Since the lemma

holds for the functions q0'(~') and q : ( r ) the corresponding measures ~t"

and F" belong to ~ ( 1 , 1) while ~'----~"== 0 and ~'~ and ~[o are positive.

By Theorem I, then, ~ ----- ~ ' ~ = 0 and d~t (I) = q:(~.)d~t" (~) + q : (~)d~'(~.),

and since the functions q~' (~) and ep" (~.) are bounded on their half axes of

regularity for I X I ~ 1, the measure ~t belongs to ~[1~(1, 1). Thus ~

exists and equals

lira qo' (i?/) q~" (i~) ----- [~'~ [ ~ > O.

Conversely, if ~t= O, ~ is in ~ ( 1 , 1) and ~o0 is positive, we can find

> 0 large enough so that ~' /" 0

+ ~./" d~t(~_~)> 0 and - ~ 4 - - - . / ' d~(~ ' )>O.


Then the functions

�9 , ( O = T + J ' + . :

belong respectively to P' and P" (by Lemmas 2 and 3) and hence the corresponding

f l and f2 belong to ~ - ( 1 ) and (~+(1) respectively. Since qo(~')--q~(~')>O

for [ < - - ~ and qg(r)--q~2 (~')>O for f > r , the comparison theorem gives

f(x) ~ f,(x) for x < - - r , f(x) ~ f2(x) for x > x ,

and thus f E ~ ( 1 , 1).

In the theorem which follows we are concerned with the estimation

of certain integrals taken over certain curves in the upper half plane. Before

this theorem is stated it is necessary to introduce some further notions.

If f(t)and g(t) are two functions defined over a certain set, they

will be said to be equivalent if there exist positive constants m and M

such that m ~ I f (t) I / I g (t) t ~ M for all t. We write f (t) ~ g (t) .

By a p a t h we shall understand the continuous image ~'(t) in the

open upper half-plane of a closed interval a _<__t< oo on the real axis where

a > O . The same path may be considered with various parameters, but we

consider only paths for which ~" (t)-->-~ for t -~ co. A path will be called

r e g u 1 a r if and only if ! r (t) I =/9 (t) increases with t and may therefore

be taken itself as a parameter, and if in the corresponding representation

= ~" (p) is a function which satisfies a Lipschitz condition. Thus there will

exist a constant Cl such that i~(p')--~.(p~)i ~_ ClIp'--p'J, or equivalently,

there will exist a positive 8 < ~ t / 2 and d > o such that for p'<p'<p'+d

[ " llArg < ~ (r) A regular path may therefore be written ]

~(p)=pe i~ For any R > O the part of the path for which [ ( l ~ R is

rectifiable, and for the differential element of arc length we have

Sometimes it will be convenient to write [(,o) = $ ( p ) + i ~ / ( p ) .

A regular path will be called n o n t a n g e n t i a l if l iminf ~ 7 ( P ) > 0 . ~--~x~ ~o

7. If C 1 is given we can take ~ and d such that cos ~,1l(4CI), d=*P0/(2C~l) where p0----min Ig(0)l- If ~ and d are given we can put C,--l/cos~.


in "this case there will exist a posittve ,~ such that the inequality

< co (p) < ~t-- E holds. A regular path is t a n g e n t i a I if lira sup ~---(p--~) = 0 ; ~ - ~ P

it will then be evident that to ~o) approaches either 0 or ~ with increasing p ;

in the former case the path is called r i g h t t a n g e n t i a l and in the

latter l e f t t a n g e n t i a l . For a right tangential path there will exist ,o,

such that for p>p~, co(p) ~ f - -8 . If then p '~p">pj , we have

.. = ~ .~ (,o') - ~ ( ,o') [ c21 r(p,)_ ( p - ) ! < ~ ( , o ' ) - ~ ( p ' ) < ,

( ~ 8 ) Thus ~(p) is an increasing function o f p for where C z = c o s ~ 2 "

sufficiently large p and ~ itself may be taken as a parameter for the

1 representation of the path. It is clear that dE ~ ! d r , ~ Cz-z d*~' and there-

fore that d8 is equivalent to the element of arc length [df i for sufficiently

large t f ! . For z ~ O a path will be said to be tangential of o r d e r 1: if

and only if there exist positive constants C3 and C4 such that

c~ p-~ _< ~ (,o)_ =<_ G p- : P

that is, ~ (p),~p'-~. A nontangential path will therefore be of order 0.

We remark that a regular path is bounded away from the real axis in any

finite part of the plane.

Let F be a regular path represented by ~== ~(,o), 1 ~ p < o o . By 1

-~-F where k > 1 we shall denote the path obtained from F by the

1 homothetic transformation ~ = [ / k . Writing -)- F in terms of the parameter

1 ,o~ = [~, i we have ,o~ = p/k, tot (p~) = to(kp,) and ~, (/9,) = --~-~(kp,).

1 It is easy to verify that - ~ F is a regular path, and that it has the same

1 Lipschitz constants Ct and C2 as F. If F is of order * then ~ F is of

the same order, however, the constants in the equivalence of ~ (pl) and

p~-r are different; we have

C3 k-" p7 ~ <= ~' (p') <= C4 k -~ ,oV~.


The following theorem, stated for right tangential paths, has an evident

analogue for left tangential paths.

Theorem III. L e t ~(~)=U(~)+iV(~) b e l o n g t o P w i t h

a = O a n d ~t t h e a s s o c i a t e d m e a s u r e ; l e t r b e a r i g h t

t a n g e n t i a l p a t h o f o r d e r z a n d y a r e a l n u m b e r in t h e

i n t e r v a l - - T < y < 2 ; t h e n t h e i n t e g r a l

(3.3) f 1 ~ i-' v ( 0 t a~ i P

is f i n i t e i f a n d o n l y i f (i) i ~'-~d~tO')< ~ a n d (ii) f o r g

1

r+.<2, j for f Lo ! 1

w h i l e t h e r e is n o s e c o n d c o n d i t i o n f o r y + ~ > 2 . 0)

Proof. Since the path F is regular, the integral (3.3) computed along

any finite segment of the path is finite. For the proof of the theorem it

may therefore be supposed that the initial point of r , ~" (a), lies on the unit

circle. In view of the equivalence of dp and Ida" i the finiteness of (3.3) is oo

equivalent to the finiteness of f ~ j " ] ~.--~(P)~(,o) ]2 d~(~)do. Since the 1

integrand is positive, the order of integration may be changed, and (3.3)

is finite if and only if the integral f F 0')d~t (~,) is finite, where

i ,~ (o) (3.4) F0.) = p ~ ] ~ - ~'(p) I' d,o.

We shall show (a) that

finiteness of f F 0.) d~ (~.)

and that (c) the finiteness of fF (~.)dbt(~) 1

1

I

j F(~.)dbt (g) is always finite, that (b) the

is equivalent to condition (ii) of the theorem

is equivalent to condition (i).

8. This theorem contains as a very special case (the case of the nontangential path formed by the imaginary axis) a theorem of I.S. Katz [8].


(a) For !~! ~ 1, !~--~'(~o) t is equivalent to p and ~(p) is bounded

by C4p *-r. Thus F(~) is bounded by the constant C4 f p-t"r-vdp and I

is therefore bt-integrable over t~. t --~ 1.

(b) For J ~ - - l , t~,--~'(/9) 1 is equivalent to p + l ? , j , since F is not

left tangential. Since ~ (p) is equivalent to pl--r we have F Q.) equivalent to

p l - - T - - y

(p+J~.{)z dp. After the change of variable p = J~it this becomes

if j" + ' c < 2,

1

then

i ll_.e_ r (1 + ty

1

p--v--r 1 (1 + t) ~ dt - [~.i~+ v- G (~ ) .

o o

dt ~ G(X) ~_ j ( 1 + 0 ~ 0

dr,

G(~.)-'is equivalent to a constant, and FQ,) is equivalent to l~l - r - v .

This establishes the first part of condition (ii) of the theorem. If y + z = 2,

G (4) is given by

the second integral

1 ( d, ] d, ,. t ( l + t ) 2 + t ( 1 + 0 2 ;

is a constant, while the first, in view of the fact that

(1 + 0 2 is equivalent to a constant in 0 ~ t ~ 1, is equivalent to 1 + Log J)'t

Log I~.] + constant. Thus F(~) is equivalent to ~z and the second

part of condition (ii) follows. If y + z > 2, G (~)'~t~[v+r-2, F (~)~ [~,! -2 and - - I

f F (4) a~ (~) < o~.

(c) For ~.> 1 we make the change of variable p = ~p~ in (3.4) and 1 1

recalling the function ~1 (pl)-------~.-r(~p,) which describes the path - ~ I

we obtain

I00 1 ~" ~2,(pt)dp, F (~) - ~,, - z~ ~, ,o~,I 1 - ~, (p-,jl 2

U).

0,48 N. ARONSZAJN and W. F. DONOGHUE, Jr.

We complete the proof of the theorem by showing that IQ.) is equivalent

to a constant for large ~., whence FQ,) is equivalent to 1/), v for such ),

and is ~t-integrable for ~ ~ 1 if and only if (i) is satisfied.

Let s be a number in the interval 0 < s < 1; it will be convenient to

choose s more particularly later. Let C, denote the intersection of the square

of side length 2s and center at ~" = 1 with the upper half-plane. The path

~ IF is next decomposed in the following way: denoting by S the set /L

Ift~2 we take

F I = ( ~ F ) f'~(S--Cs), F2=(~F)--(SUCs) and F 3 = ( ~ - Y ) ~ C s .

11 (X), 1~ (Z) and 13 (2t) will denote the corresponding parts of the integral I(~.).

On the set F1 it is clear that ! ~1(pl)-- 1 1 > s , whence 1, (Z) is 2

bounded by ~T ~ol-Vdpl, since ~ , (p l )~ ,o j . If z = O then ~l(p,)~mp, O

2 for some positive m and It (~.) will be bounded from below for ~. > 1------s

1 ~ $

m

by 9 , f ,o~-~ a,o, . (1---s)12

On the set F2 we have t~'1(pl)-- 112_ -~*.r i / 2 and therefore 12(J~) is o~

bounded by C~ ~.-~ 4 j ,o~ -*-r-v dp,. Thus I~ (Z) + 12 (h) is bounded in ~.,

2

and if �9 this function is bounded from below, and therefore equivalent

to a constant. However, if x = O the number s may be chosen so small

that the set F3 is empty for sufficiently large Z, and therefore IQ,) is

equivalent to a constant.

The proof of the theorem is completed, then, under the hypothesis

that ~>O, and it will be shown that [a(~) is equivalent to a constant.

Since o (p ) converges to O with increasing p it follows that for sufficiently

large ~. the set F3 is uniformly close to the real axis; F3 is then a connected

set, in fact, a path segment. Moreover, for sufficiently large ~. the coordinate

$'~ may be used as a parameter instead of p , .

In the rest of the proof we will omit the subscript, since all integrals

are taken o v e r F 3 .


Since pv is equivalent

equivalent to the integral

to a constant over I'3 the function I~Q.) is

l + s

J (;t) = (~ - 1)2 + n (~J~'- '

the dependence of which on ~. takes place through the dependence of the

1 curve - ~ - r on L For any two points on r3 we have

C3 Z -* p,t-~ <. n (p,) ~ C4 ~.-" p , t - , ,

and therefore

whence

,~ (,o") - ~ \ ~ - 1 ~ ~ ' i - ; _<_ m < ~ ,

< - - < M M = n (P ' ) -- "

An upper bound for JQ.) is found by setting H(~.) and hQ.)

respectively equal to the maximum and minimum of ~(~) on the interval

t - - s ~ . _ < = l + s , and writing

I + s

j ( ~ - 1) 2 + h(~)~ = ~-1iT 2 arctan \ I - ~ T -~ M 2 , . I - -S

A lower bound is found similarly:

1 + s

�9 (Z 1)2 + H~-~.)-2 = - - - ~ H 2 arctan 2__ - ~ arctan ,

and for large ~., H(~.) converges to 0, and therefore this lower bound is

bounded away from 0.

L e m m a 5. (a) L e t {p"(f) be a f u n c t i o n in P" a n d k be ~ 1 ;

t h e n i f (x ) b e l o n g s t o @+(k) i f a n d o n l y i f Ft" b e l o n g s t o

9J/+(k) a n d ~"~ > O.

(b) L e t (p'(~') be a f u n c t i o n in P ' a n d h be ~ 1 ; t h e n

i f(x) b e l o n g s t o @-(h ) i f a n d o n l y i f ~t' is in ~O~-(h), a ' = O

a n d ~'~ > O.


Proof . In view of Lemma 4 it is enough to prove that if all conditions

in each case of the present lemma are satisfied with k = 1 (or h -- 1) then

f" ~ ~+ (k) for k > 1 is equivalent to ~" ~ 9Y~ + (k) (or similar statement for

case (b)).

Case (a). We first consider an auxiliary conformal mapping.

The function ~ - f ( z ) = z + 1/z maps the exterior o f the unit circle

in the upper half z-plane onto the upper half f -p lane; the inverse of this

mapping, denoted by Z (~') is therefore in P. For p > 0 we form the function

~" = Zp (~') = [Log (e 2p Z (~'))]P, and it is easily verified that Xp ( f ) is also

in P, in fact, this function maps conformally the upper half plane onto a

region in the first quadrant of the r , plane, and carries the semi-axis ~ ' > 2

onto the semi-axis ~ " ~ (2p) p. W e let r p denote the image of the imaginary

axis: f ' ( ~ ) = zp(ir~), ~ ~ I, and will show that this is a right tangential

path of order 1/p. Making use of the estimate Z(irl)=i~4-O(-~)i t is

easy to find that

~,(77)_~(Log~),[l +p 4p+oti ( 1 ) ] 2 Log ~ + O (Log ~)2

and the determination of the order follows from this estimate. W e also

find that ! d~'~ I~, is equivalent to P (L~ and that I~"(~)] is equivalent d~ ~

to (Log~) p .

Consider now the function q~*(~)=q~"(Xp(~')) . This function is in

class P. Furthermore, if q~*(i~2) converges to a limit (finite or infinite)

for ~2 o - ~ this limit must be == ~3~. In fact this would be the limit o f

q~" (~") along the path Fp and since fp" (~") converges to [3~o along the

imaginary axis the limit along Fp must be the same.(9) I t follows that

r~*-~ 0 (otherwise q~* ( iT )o . o~) and that if the corresponding measure ~t* S - - belongs to 9X(1 , 1) then ~30~- [3~

1 For k > 1, choose p = k--~l ; the measure IA," will belong to ~[Y~+ (k)

9. By a mapping of the upper half-plane onto the unit circle we transform the function ~0" into a bounded function and then apply the well known theorem cf identity of limit values of a bounded function along two paths converging to the same boundary point (see for instance [13]).

ON EXPONENTIAL REPRESENTATIONS...

00

if and only if the integral ~" ~-2d~t" (~.) is finite, and this will take place

1

if and only if the integral . f !~"[k'2 Im [q~"(f')] Ida' is finite, in view of

1"/,

Theorem III. Writing this integral in terms of the parameter ~, 1 ~ 7 < ~ , 0o

we find that ~t" belongs to 9J'~+(k) if and only if /'~-t lm [q~* (i~)] dTl t

is finite. Again by Theorem lII (which can be applied since o*----O) this

integral is finite if and only if V* belongs to 7Yt(1,1).

The same argument, applied to the function Log eP(~'), shows that

f (x) is in @+ (k) if and only if if(x), associated with ~D*(~'), is in @(1,1).

If i f(x) is in @(1,1) , Lemma 4 shows that Ix* is in ~[R(1 , 1),

while if p* is in ~ ( 1 , 1) then by our previous remark ct*=O and

[~*~= [ ~ > 0; hence Lemma 4 puts if(x) in @(1,1) . Thus (a) is proved.

C a s e (b). The argument parallels that of ( a ) w i t h the auxiliary

conformal mapping,

~' = ~ (O -= - [ - ~ i + Log (e~P Z (O)F-

L e m m a 6. I f if(x) c o r r e s p o n d s t o ep*(r) in P, t h e n

(a) f*(x) b e l o n g s t o @+(*) i f a n d o n l y i f ct*=[3~--0 a n d

Vt* b e l o n g s t o 99l+(1)--~0't+(2).

(b) if(x) b e l o n g s to @-(*) i f a n d o n l y i f c t ~ ~t* is in 0

, f a~* (z) sJJ~-(O) -- 9J~-(1) a n d ep*(O) --- ~31 + jL(~.2+I ) ~ O.

Proof. (a) By Lemma 3, Section 2, particularly (b), tt ~ 13 * = 0

and ~t* belongs to ~lY~ + (1); this measure cannot belong to ~[R+(2) since in

that case

-;c0" (O = a~* (~) - ~ _ ~

0 0

= exp ~i + , x--~" x2+1 (f~ (x) -- 2 + (x)) dx 0

and passing to the limit as ~ approaches - - ~ we would have


o~ oo

If l [ dl x" (~.) = exp (o~) exp --x (f* (x) -- X + (x)) dx < XZ+ 1

0 0

which would put f* (x)--'Z+ (x) into @+ (I), contrary to hypothesis. Since

the last argument is reversible, the proof of (a) is complete. Part (b) of

the lemma immediately follows from Lemma 2, Section 2, particularly

parts (b), (c), (d) and (2.5).

Occasionally in our proofs we shall have to consider functions ~p (~')

regular for I m ~ > O and with n e g a t i v e i m a g i n a r y p a r t s Imap( r )<o .

This class of functions will be denoted by Q. Their theory is completely

parallel to the one for functions in t'. We have a canonical correspondence

between functions r in P and ~p in Q:

( 3 . 5 ) ,P ( O = (--s , �9 (O - - , ( - O .

All the representations of q9 lead to parallel representations of q,

with ct replaced by --c~ (hence ~ 0), I~ or ~ unchanged and the measure

dlx(~.) replaced by dv (~.), v (~.)= Ix(--).) (hence dv ___ 0). In the exponential

representation cr~ is unchanged and f ( x ) is replaced by g ( x ) = - - f ( - - x )

(hence --1 ~ g(x) <= 0).

We introduce subclasses Q" and Q" of Q similar to P" and P" (i.e.

by the conditions g ( x ) = O for x>O or x<O respectively). P' and P"

are transformed by (3.5) onto Q" and Q" respectively. Furthermore, by

(3.5) measures Ix in ~1~(h,k), ~ - ( h ) and ~ + ( k ) are transformed into

measures v in ~[l[(k, h), 9J~+(h) and 9Jl-(k) respectively. Also functions

f in @~(h, k), {~-(h), 1~* (k), {~-(*) and @+(*) transform into functions

g in @(k, h), 1~+ (h), {~-(k), {~+ (') and •-(*) respectively.

By virtue of these correspondences we can translate every statement

about functions in P into a statement about functions in Q; the last

statement will be denoted in the same way as the one for P. For example,

Lemma 2, (w 2), when applied to Q, concerns functions in Q ' ; condition

(2.5) becomes now

or)

j " dr" (~.) ,l," ( 0 ) : 4- _ 0

0

statement (b) of the lemma becomes: ~t"<~O if and only if


x [g"~x) + z + (x)] ax < OQ

xZ+ t 0

Before turning to the. proof of Theorem A, we note two useful

statements which are easily proved by direct computation if one keeps in

mind Lemmas 2 and 3 for Q" and Q' respectively:

1) If ~P'~ Q" is representable by 0

f d%(~) ,'(C) ~ t + ~_~

then q: (~') = ~'~p' (~') E P' and is representable by 0

2) If lp 'E Q" is representable by

�9 dr" (z) k - - ~

0

with v" E~0~ + (1) then q~" = -- ~-xp" (~')E P" and is represented by

o0 oo

. , �9 +

0 0

Proof of Theorem A. Our method of proof will depend heavily on

Theorem I and the factorization of q~ (~') into a factor in P ' and one in P ' .

We recall that this factorization is unique up to an arbitrary multiplicative

constant, and in the computations which follow we will generally fix that

constant, although often its determination is not necessary for the argument.

Throughout the proof the decomposition of tV will be given by

~V -= ~ (P', qP'~ P', q~" ~ P ' - The functions f ' ( x ) and f " (x) are given by

f ' (x) = f (x). Z- (x), f " (x) = f (x). X + (x).

By dlx + (~.) we shall mean the measure ~t restricted to the closed right

half-axis, and by dlx-(~.) the restriction of !~ to the open left half-axis.

To explain the arrangement of the proof let us denote by A (h , k),

B (h , k), C (h, k) and D (h , k) the subclasses of P of functions satisfying

the condition imposed on f in the cases (a), (b), (c) and (d) of Theorem A


respectively. Similarly, AI (h , k), B1 (h, k), Ct (h , k) and Dt (h , k) will be

composed of functions tp in P satisfying the other conditions in (a), (b),

(c) and (d) respectively. The theorem states that the corresponding classes

with and without subscript are equal. We note then that each class with

h = k = = l contains all those of the same kind with h>_l and k > l .

Furthermore, A (1 ,1) , B ( 1 , t), C ( 1 , 1 ) and D(1 , i) are mutually disj~

the same as A t ( I , t), Bt (1 ,1) , C t ( 1 , 1 ) and D I ( I , 1).

In the first part of the proof we will show that

A (h , k) --- At (h , k) f~ A (t , t)

and similar relations for the classes B, C, and D. This will give in particular

A (1 ,1 ) C A I ( 1 , 1 ) , ..., D ( I , 1) C D r ( I , 1). In the second part we will

show that if q~ does not belong to any of A ( 1 , 1 ) . . . . . D(1 , 1) then it

does not belong to any of A1 (1, t) . . . . . Dt (1 , 1). This will give

A ( 1 , t ) = A t ( 1 , 1 ) . . . . . D ( I , 1)-- - -Dr( l , 1)

and, in view of the results of the first part, the complete statement of

the theorem.

We note first the exponential

functions which are in Q: 0

and

representations for the following two

1 x ] } x--g x2+t . dx

--~ = exp -- x - ~ " x2+l " o

F i r s t p a r r o f p r o o f .

(a) Let f ( x ) - - Z - ( x ) belong to • (1 ,1 ) . We may therefore write

s( ) q~ (~') = exp [oi + 1 x . - - ~ x--~" x2+ 1 I f (x) -- Z- (x)] dx]

= exp (*| exp f f (x) -- Z- (x) dx = exp (~=) ~p' (~') ~p" (~')

where lp'E Q', 0 o

{ * f ' ( X ) - ~--(X) ~ dv' (~.) ~p'(~') = exp j - - ~ dx = 1 +

x--~. ~! t - -~ woo m00

- - - - , dr ' ~ O,


and q)" (~ P " ,

f dr" (a.) q f f ( ~ ' ) = e x p f " (X) d x = 1 + dv " ~ 0 . x - ~ ~ - - ~ ' -

0 0

(The representations of ~p'(~') and ~p"(~') in terms of v' and v" follow

from Lemma 5.) For the factorization of q~(~) we have then

r (~') = exp (~| ~ ' , ' (~) and (p" = ~p".

By (3.6) and Theorem I we have

r = ~' f3~ = exp (*~o) > 0

a ~ - (a.) = ,~" (a.) a~ ' (x) = , " (a.) ~x ~v' (x ) , a. < o ,

and

d~+ 0,) = *' (a.) dr," (a.) = aa.~p' Ca.) dv" (a.), a. ~ o .

Thus 9 - will belong to ~ t - ( h - - 1 ) if and only if v' belongs to ~3~-(h),

and this will happen if and only if f ' ( x ) - - Z - ( x ) belongs to {~-(h) , by

Lemma 5. Similarly, ~t + will belong to ~[R + (h- - l ) if and only if v* is in

~ + ( k ) , which happens only when f* (x) belongs to @+(k). Thus

f ( x ) - - X - ( x ) in @ ( h , k ) is equivalent to It being in ~ t (h - -1 , k - - l ) .

(b) We assume r E @ ( 1 , I ) .

r (~') ---- exp (~s~) q / ( r ) ~p- (.~), 0 0

f f C.',:) f a~,'ca.) av'~ o t P ' ( O = exp d ~ _ ~ d x = 1 + ~ _ ~ , _ , ~ 0 0

OD OD

, " (~') = exp (x) _ l + dr" ~ 0 x - - ; " . a . - -~ ' - "

0 o

q~' (~') --- exp (o~) ~p' (~'), q~" (r) = ip" (~'), r = 0, ~3~ -- exp (o~) > o .

d V - (t) = q," Ca.) a~" Ca.) = *" Ct) ~ aY' (X),

I~- in ~ - ( h ) is equivalent to f ' (x) in @ - ( h ) ,

~+ in ~ER + (k) is equivalent to f " (x) in @+ (k) .

(c) We assume f - - 1 ~ @(1 , l ) . We could proceed in the same way

as for the other cases but it is simpler to notice that the reflection

( 0 = - �9 C----~)

(see C2.1)) transforms the case (c) into the case (b).


(d) We assume f - - z + ~ @ ( 1 , 1 ) .

- ~ (~') = exp (T | (r 'L'" ( 0 , 0 0

q/ (~) = e x p - - - - ~ d x = 1 + ~ - - ~ , dv '> O

0D 0o

f " (x) - - Z + (x) dx = t + - - - - # , dr" < o ap" (~') = exp. x - - [ ~, - - ~ = " 0 0

M = exp (%~), ,p ' ( f ) == M v ' ( f ) , ,p" ( f ) ----- -- ~p" (~') / f .

By (3.7) and Theorem I, a = O, [3=---- O, [~oo = [:~'~ " [3oo = O,

d ~ - (~.) = ~p" (~.) d[~' (k) = -- -~- (~,) Mdv' (~.) for ~, < O,

?- d~ + (~,) = r (~) dg" (~,) = M~p' 0,)- - dv ~ (),) for }~ > 0 .

~ - in ~ ' - ( h + 1) is equivalent to f ' ( x ) in @ - ( h ) , and similarly l~ + in

~ [ R + ( k + l ) is equivalent to f* ( x ) - - X+ (x ) in (~+(k). It is easy to verify

that M = total mass of ~.

S e c o n d p a r t o f p r o o f . We assume that q~ does not belong to

any o f the classes A ( 1 , 1 ) . . . . , D ( I , 1) , i .e. that f does not satisfy any

of the conditions in (a) . . . . . (d) with h = k = 1. This is equivalent to at

least one of the two formulas f ' E @ - ( ~ or f ' E @+(*) being true.

1 ~ If f ' E ( ~ - ( * ) then, by Theorem I and Lemma 6, a = c t ' ~ o = 0 ,

hence q~ ~ At ( 1 , 1 ) . Lemma 4 gives q0 ~ B, ( 1 , 1 ) . Reflecting the function q~

(see (2.1)) we obtain 4"~ with f ' E ( ~ + ( * ) , hence by Lemma 4, ~ / / B t ( 1 , 1 )

and q ~ C t ( 1 , 1 ) . Finally, Lemma 6 gives t t ' E g J ' ~ - ( 0 ) - . ~ R - ( 1 ) and

Lemma 3(c) shows that lira inf(--~.tp" (~.)) > 0 . Since ~-->--O0

d ~ - (~) = q~- (~) d~' (x ) ,

~ * - ~ t - ( 2 ) and so q ~ D t ( 1 , 1 ) .

2 ~ . I f f " ~ @+(*), the reflected function r will have f ' ~ @ - ( ~

The above proof shows ~ At ( 1 , 1 ) tJ ... tJ Di ( 1 , 1 ) and hence the same

is true for q~.

Theorem A treated functions r in P for which neither f ' = f x - ~ (~- (*)

nor f ' - - - - - fx + ~ {~+ (~ holds. Our second main theorem will characterize by

properties of the measure ~t those ~ in P for which one of f ' and f "


belongs to the respective @+ ( ' ) and the other not. These properties o f rt

will display a relation between r t - and ~t + (in distinction to those of

Theorem A which decomposed in separate properties of r t - and ~+).

In order to express these relations we have to introduce some notations.

It should be remembered that since the functions q0 under consideration do

not belong to any of the cases of Theorem A, we will have: 1) a = O;

2) if r t E ~ 0 ~ ( 1 , 1 ) , ] ] a o : 0 ; 3) r t ~ J ~ ( 2 , 2 ) .

B. 1). r t - and ~+ will be essentially (as before) the restrictions of rt

to the negative and positive half-axis except that if rt [ 0 ] > 0 , we will now 1

put the mass y ~ [ O ] at 0 in r t - as well as in rt+.(t~ The class ~J~-(h)

will now be extended to include measures with support on 4_< 0.

B. 2). I f ~lE~rJ~(1, l ) (hence ~oo :: 0), we put

(3.8) q0 = ~ - + q,+ ~ - = ~ + = - - - ~ .

~ 0

B. 3). I f r t * ~ ( 1 , 1 ) , we put

0

(3.9) (p = q0-- + q0 + , q0-- 2 , 4--.~" 7,2+ 1

O0

j ' ,~+ = lj_t + [ ] art+ (4). 2

0

It should be noticed that our definitions were chosen in such a way

that for the reflected function ~ ( ~ ' ) - - q~(--~) the corresponding measures

and decomposit ions are given by reflecting rt+, r t - , cp+ and q ' - .

Before stating Theorem B we make a last convention. For a measure

d(o and a positive function ~9 (4) defined for )~ > 0 (or 4 < 0) we will say

that O(4)d0o(4)E~0l+(k) (or ~X-(h) ) f o r l a r g e t4}, if the measure

,9 (4) de0 (4) restricted to some infinite interval 4 > A > 0 (or 4 ~ - - A < 0)

belongs to !IX + (k) (or ~1l- (h)) .

T h e o r e m B. F o r h a n d k > l a n d f u n c t i o n s *p(~) in P

c o r r e s p o n d i n g t o f ( x ) = f ' ( x ) + f ' ( x ) w e h a v e

10. As a consequence of this slight change, formulas ar t - (4) = ~0" (4) art" (4), art+ (4) = ~" (4) art" (4)

in Theorem I may cease to hold for k=0.


(a) f'(x) is in ~J-(*) and if(x) in {~+(k) i f and o n l y i f d~+ (~)

a---O, bt- is in ~R- (O) -- ~[R- (1) and lq0_(~.) ! b e l o n g s to ~/+(k)

f o r l a r g e IJtl;

(b) f ' (x ) is in ~ - (* ) and f " ( x ) - ~ + ( x ) is in {~+(k) i f and dr + (~.)

o n l y i f a = O , ~t- is in ]B/ - (1) - -~r / - (2) , -iq_(/t) I is in ~/+(k)

f o r l a r g e 1)~], and ~30o= O;

(c) if(x) is in @+(*) and f'(x)--X-(x) is in @-(h) i f and d r - (x)

o n l y i f c~=O, ~+ is in ~/+ (0) -- ~ + (1) and !q~+()01 b e l o n g s to

~3/-(h) f o r l a r g e 1~1;

(d) f '(x) is in @+(') and f'(x) is in @-(h) i f and o n l y i f

d~t-(~) is in ~ - ( h ) f o r l a r g e {z=O, ~t + is in ~/+(1)--~IR +(2), Iq~+(Jk)!

Ixt and ~ = 0 . In the proof of Theorem B we will refer to two simple lemmas

which follow. Lemmv 7. (a) I f ~tE~/--(O)~[F/--(1) t h e n t h e f u n c t i o n

0

'~ (.0 = I~ + ~._~ ~ + ~

s a t i s f i e s f o r ~ ' > 0 : q~(~)-~-e~ and --~-~)--~0 w h e n ~ ' ~ + ~ .

(b) I f ~EgJ' t-(1)--~Yt-(2) t h e n t h e f u n c t i o n 0

f a~(~) ~ 0 0

~ > : q~l(~')<0 and q~, ( .~)- ->-0 , - -~ 'qh(~ ' )~+~ s a t i s f i e s f o r

w h e n ~" ->. + eo . o

f a~(~) (c) If I~E~[Y~-(2) t h e n t h e f u n c t i o n q~1(~') = k - - ~ n 0 0

0

s a t i s f i e s f o r ~ '>0: - - ~ ' q h ( ~ ' ) ~ --= fa~,(~) w h e n ~ ' ~ + e ~ .

Proof. (a)The function under the integral defining tp increases for

-->. + oD to ~2+1 , hence the first assertion. Since

ON EXPONENTIAL REPREgENTATIONS... 3~9

0

q~ ~+ f 1 4- ~[ dv (~.) and 1 + X~"

is uniformly bounded and ->-0 for ~"-->-oo the second assertion follows.

(b) The first assertion is obvious, the second and third follow from

(a) if we write 0 0

- ~ , ( 0 = ~ [o1 + ~ - c ~ ( - ~ ) ~ ( ~ ) = ~ + ~ - - 0 0 ~ 0

0 j ' [ ' * ~.--C" ~.~+1 (--~') d~t (~.),

~ 0 0 y,

(c) follows from the fact that ~ is uniformly bounded and ~ 1

for ~" + + o o .

Lemma 8. Le t $(~) be a c o n t i n u o u s n o n n e g a t i v e

f u n c t i o n f o r X ~ 0 and l e t l i m tg(~)--#(- -oo) , O < 0 ( - - o ~ ) < c ~ . ~,--0. - - 0 0

(a) I f q~EP h a s a m e a s u r e ~ E ~ - - ( 0 ) - - R R - ( 1 ) and ~p has

a m e a s u r e dv (~)= O(k)dH(k) t h e n l i m qJ(~) ----0(--oo).

(b) I f ~ t ~ 0 ~ - ( 1 ) - - ~ - ( 2 ) and 0 0

�9 , ( 0 = d ~ (~) ,~, ( 0 = k - - t " ' ~ - - { '

~ 0 0 ~ 0 0

, , (0 t h e n l i ra = ~ ( - - ~ ) . ~-++| *~ (-0

Proof. We prove first (b). The assertion follows from

~7" ~ T 0

2 ** (0 2 A~ (T) f a~ O0

where T is a fixed arbitrarily large positive number, A, (~)= inf 0(~), x < - ~

A.,(~) = j~p_ o(~) and C>O. We use here temma 7, (b) a,d (0, to

show that

lira 1 f d____~(~._)) = 1, lira 1 ; 0(~)d~t(~______)) _-- 0 so+| q~* ( 0 ~ - - ~" ~-,+| ~* ( O - t - [ "


To prove (a) we first notice that both *~ and *p belong to case (a)

of Lemma 7. Therefore there exists B > 0 such that q0 (~') and ap (~') are

positive for ; " > B . The translated functions q ~ ( ~ + B ) and ap(~ '+B) 1

are in P' and their product by -~-~ is in P. A simple computation

shows then (we could also use Theorem I) that

- - B

(7 = ~ a~t(~. + B) + _~. - f , ~ 4.8) ~ ~ - --QO

- - B

1 - - 7 q ~ ( ~ + B ) = - - O(~+B)a,aCZ+B) + v(B____)) - - 0 0

and these two functions are in case (b) of the present lemma, proved above.

Remark. By reflection we obtain from Lemmas 7 and 8 parallel

statements concerning measures in the classes ~0~ + . We will refer to these

by saying, for example, "reflected Lemma 7, (a)".

Proof of T h e o r e m B. It will be enough to prove cases (a) and (b)

since (c) and (d) reduce by reflection to (a) and (b) respectively.

N e c e s s i t y o f (a). From Lemma 6 and Theorem A, the factorization

of q~(~') leads to functions cp'(f) and q~"(~') for which r 0, f is in

9X- (0) - - ~ - (1), I ~ > 0 and ll" in ~ER+(k). Thus c t = 0 . Since the factor m

'V" Q') occuring in d ~ t - ( ~ ) = q~"(1)dp'Q,) converges at infinity to ~o~ > 0 ,

~t- is in the same class as ~'. We may write

0

f l 1 - 'P-- (;') = ~ _ ~- ),2 + 1 qo" (~,) dp.' (J~) + ~ - [0] + ~ i . --~" 2 '

by Lemma 8(a), we get

lira q'-(~') =13~o. g ~ + ~ qr

Now [a" is in OrJ~ + ( k ) , and dlx'Q.) may by written

for large values of J~. Thus

q~- (x) a~+ (x)

~ + (k), for large I~.1.

d~+ (~)

at least 9"(~) ~o- (~)

is in the same class as I~', i.e. in


N e c e s s i t y o f (b). Arguing as before from Lemma 6 and Theorem A,

we find that the factorization 0f~9 ( f ) leads to c (=0 , lg' in ~[Yt-(0)--~'-(1),

~ o = 0 and ~t" in ~ + ( k + t ) . Thus c t = O and d~t+(~.)=qo'0.)d~t"(~.) is

in 7J~ + (k) (since *p'(~,)= o (~.), see Lemma 7, (a)). From

a~,- (~) = q0- (~) aft (~)

we infer that ~-- is in ~l~-(1)--~J~--(2), in view of the fact that It" has

finite total mass ~t~ and therefore that q~*(~) behaves asymptotically like

_~. (see reflected Lemma 7, (c)). Now we have ~t in ~J~(1, 1) and ~

exists; this number however is equal to

lira q~ (~') -~ lira go' ( ~ / .." lim ~'q~" (~') = -- ct' po = 0 g.-->-oo g-r g-->-w

as f approaches infinity in any angle.

1 The function ---~-~9' (r) is in P and is represented by

0

- ~ q~ ( .0 = - a . ' (a) + qo'(o__.___2)

where o < q ~ ' ( O ) < o o (see Lemma 2, (2.5)). Comparing with

we see that

by Lemma 8, (b), lira g->-'t" o,z

It follows that

d~+ (~)

for large I~.t.

0

, , , - ( o = , t.- T)' d.' + - 00

( - - ~ ) d ~ t ' ( ~ . ) E ~ - ( l ) - - ~ - ( 2 ) , - - ~ . q J " ( ~ . ) - - > - p~ and hence \ l

- ~'q~- ( 0 at

+ ~ o " ~0' (.0

S u f f i c i e n c y o f (a). Since t t - ~ C R - ( O ) - - ~ - ( 1 ) , q~-(~.)--~-+~

for ~ , - ~ + o o and hence for ~ , > A > 0 , q ~ - ( J ( ) > 0 . The measure

d0~ (~) = dp.+ (~) q~- (~) '

~.~A is in ~ + ( k ) and we can choose B > A large enough so that

+'(~) ~a~" (~) ~ ~+ (k) I~P -(~)r


We form the function

Or)

f do 0-) I ~ - - A < T "

B

1 i'" do (L) Q(C') = T + .J ;~ -c" "

B

The translated functions q~- (~" + A) and Q (~" + A) are obviously in

P' and P" respectively. Hence we can apply Theorem I and obtain that the

function q~-(~')Q (U) is in P and its measure v is given by

d v ( X ) = Q ( t ) d ~ - ( ~ . ) for ~.<0, d v ( ~ . ) = 0 for 0 < ~ , - B .

We have therefore 0

(3.10) q~(C) --q)--(C')Q(C)= C + ~.--C" ~.2+ 1 ---0el

B

0

where C is the difference of ~i for q~ and for q~-Q.

Since for ~ < 0 , O0

i i " do I Q(~)c~(A)=q- +a ~ < T ' B

the first integral in (3.10) for ~'-~+oo behaves like ~ ep-(~') (see Lemma 8, (a)),

and hence -> + o % whereas the second is bounded. It follows that for large

positive ~', q~ -- W- Q > 0, and by the comparison theorem f " (x) < g (x)

for large x >0 , where g(x) is the function in the exponential representation

of q~-Q. But this function, for x>A, coincides with the one corresponding

to Q and by Theorem A(b) the last function is in @+(k). Hence

f"E•+(k). It is clear also that f i E @ - ( ~ (otherwise q~ would be in one

of the cases (a) or (b) of Theorem A).

S u f f i c i e n c y o f (b). The proof is similar but slightly simpler

than in case (a). We notice first that q)- and (p+ are given by (3.8) and

that q~- (() < 0 for C" > 0 and hence -- C'q)- (C') ~ P' and is given by


0

Z-~" X - -oO

The measure d o (~.) = ~. t q0- ().) I restricted to ~. > B > 0

�9 "l + (k + 1) and we choose B large enough to have

j . 1 ao (~.) < -~-- B

The function

is in

GO

f do = f~ (0 �9 ~.--~" B

has therefore the property that it is in P ' , and for the function g" (x) in

its exponential representation we have X + - - g" ~ @+ (k) (by Theorem A (d))

and further o0

- - ~'{2 (~') < -~-1 for ~" ~_ 0 and - - ~'Q (~') -->- f do (~.) for ~" --~ - - ~

B

The function ( - - ~'r (~')) Q (~)

corresponds to a measure v given by

dv (~) = Q (~,) ( - - 2td~- (~)) for

is then in P and by Theorem I

2t<O, d v ( ~ ) = 0 for O < ~ . - ~ in angle and this is = 0 (since ep- (~') -->- 0)-

Hence we can write

- &o- (0 Q ( 0 =

o ~

f -- ~.Q 0.) d~- (z) Y-~-3 + f at~+O') - - ~ B

0

�9 ( 0 - ( - ~ ' ~ - ( 0 ~ ( 0 ) = j " O. + zQ (z))a~-(~) V2--g

Th~ first integral behaves like

B

B

+ j " dt ~+ 03 rz_? o

364 N. ARONSZAJN and W. F. DONOGHUE, jr.

(see Lemma 8, (b)), hence its product with --.~ converges to + ~ for

~ ' - ~ + o o (see Lemma 7, (b)). The second integral is O / -~ - ) . Thus a ~

~ / q0--(--~q~--~) is negative for large positive ~, therefore

f " (x) ~ g" (x) for large x, 1 - - f " ( x ) < = 1 - - g ' ( x ) for large x

and X + - f * E I~+ (k). As before f" must be in @-(*) (otherwise q0 would

be in case (c) or (d) of Theorem A).

4. Quantitative relations.

In this section we give certain quantitative relations associated with

the qualitative relations which form the content of Theorem A. For this

purpose we introduce the concept of moment and absolute moment of a

measure (in general, a signed measure).

For any r e a l k > - - 2 , the a b s o l u t e m o m e n t i~*!~ of a measure

tx is

(4.1) ! tk----flxt laix( )l for k_o, I for

For any i n t e g e r k ~ - - 2 , the m o m e n t Ixk is

f Zk+2 (4.2) ~tk-=-- Zkd,tt(Z) for k > 0 , t , ~ = j ~ - ~ a ~ ( x ) for - 2 < = k < 0 .

Similariy we introduce the moments fk and the absolute moments tfi~

for a function f(x), - - ~ < x < ~ , by using the corresponding measure fdx . A moment Ixk will have a meaning if and only if I~*lk< oo.

We note the obvious relation for integral k: 1 ~ if support of IX is

on ~ 0 , Ix,=ilxlk; 2 ~ if support of ~* is on X_<_O, Ixk=(--1)klbtl~;

3 ~ if k is even, ~tk----tixfP..

We write IX----Ix-+ IX+ with the convention that the mass at 0 is

assigned totally to IX- if it is < 0 , to t.t + if it is > 0 (this convention

is invariant by correspondence (3.5) but not by reflection). We introduce

the mixed absolute moments for h > - 2, k ~ - 2, by

(4.3) t Ix I~.~ = I ~ - l~ + I Ix+ I~ Similarly for f = f" + f", we put [fl~,,t, = !f'lh + If"l*, with

f ' = f z - , f " = fx +. Obviously IxE~J't(h,k) means that IIxlh_2.k_2<oo.

The content of Theorem A can now be expressed by saying that

IIxlk,k< oo if and only if a corresponding function in @ ( 0 , 0 ) (depending


on the case : f -- X - , f , f - - 1 , f - - Z+) has a finite corresponding absolute

moment. If the correspondence between the function in @ (0 ,0 ) and the

measure 8 were linear, this fact would automatically lead to the result that

one moment is bounded above and below by the other moment times

constants independent of ~. In the present situation we can find in almost

all cases evaluations of an absolute moment of !t (or the function in

@ (0 ,0 ) ) by expressions involving the corresponding moment of the function

in @(0 ,0 ) (or p.) and often some moments of lower order of the last

function (or Vt). In this connection it is useful to note the simple inequalities

(4.4) for - - 2 < k < / , we have !~lk <= 21~1-2 + 1~[~, 01)

obtainable immediately from (4.1). By using (4.4) it is always possible, if

desired, to replace an evaluating expression by another containing only two

moments: one of order - -2 , and the one of highest order.

We shall indicate how such evaluations can be obtained for arbitrary

integral orders. As illustration we shall give explicit evaluations for moments

of lowest integral orders in each case of Theorem A. We shall not

concern ourselves with evaluations of moments of non-integral order (they

would require a quantitative companion to Theorem III).

Before dealing with the evaluations we shall note a few relations

between the elements of the two representations of a function q~ E P not

described by Theorem A, i.e. such that f ~ @ ( 0 , 0 ) - @(1 , 1), a----- 0,

~ * ~ ( 2 , 2 ) and if ~ ( 1 , 1 ) then t300=0. By putting in (1.4) and

(1.10) ~" = i we get

(4.5) [3i = e~ ~-2 = i~i-2 = eOi sin ( ] - z ) �9

I f ~tE~]~(1,1), ~Qo=~ 0 we get from (1.6) and (1.10)

(4.5') V-t = e ~ ( f -2 ) , :1~I-2~ * = e~ sin ( f -2) �9

We cannot have here in general any evaluation for iVl-t. However,

if q~EP*, j~ti_t is finite (Lemma 3) and given by (4.5') (note that q~ cannot

be in P" here since by Lemma 2 it would be in case (b) of Theorem A).

In what follows we will have to use some classical considerations

concerning asymptotic expansions of the form E as ~'-i + ~---k Ak (~) with 1-------0

11. Other inequalities between these tbree moments could be obtained by H61der's inequality.

3 6 6 N. ARONSZAJN and W. F. DONOGHUE, Jr.

A ~ ( O ~ 0 for ~-->-oo in any angle. We note first that if i ~ l k < ~ (i.e.

t t ~ ( k + 2 , k + 2 ) ) with k>~0, we get such an expansion by writing

dvt (1) ~ ~---t - - k - , (4.6) ~ ~ _ ~. -" ~,- , + -. ~. _ ~.

/ = 1

Furthermore, if we have two such asymptotic expansions connected

by the formula k k

(4.7) 1 - - aj ~--4 ~. Ak (5) = exp b: ~ - t + ( - k Bk (~) 1 1

then, by expanding the exponential on the right side or by expanding the

logarithm of the left side, we get

(4.8) - - aj = bi + pi (b, . . . . . b j _ , ) , bt = - - a t - - q t ( a , . . . . . a , _ , )

Here pj and q, are well determined polynomials in variables (bt . . . . . b j_ 0

or (at . . . . . at--0 respectively, with rational positive coefficients. For the

smallest indices we have: 1 2 1 3 1 --2 1 --2 1 4 . (4 .8) p,-----0, P 2 = ~ b , , P 3 = b t b 2 + - ~ b , , p ~ = b , b z + y o 2 + T 0 , b 2 + ~ b ,

1 2 1 3 l 2 2 1 4 (4.8") q t = 0 , qx = Ta t , qs -~- at a2 + Tal , q4 = al as +-~a2 + at a2 + - ~ a t �9

The next lemma will give a much more precise estimate o f Iflk in

terms of !fit for l > k than the one provided by (4.4).

L e m m a 9. L e t k a n d l b e i n t e g e r s , - - 2 < k < l .

(a) I f f c o r r e s p o n d s t o tp i n P ' o r P ' , t h e n 1 ~ I f l - 2 ~ 7 ~ - ;

2 ~ f o r k = --2, l = --1, i l l-2 ~-- arctg V-e 21ft-t - i ;

3 ~ f o r k--- - - 2 , l ~ 0, f ] -2 < arctg[([f[t(14-1))t/~'+t)]; 1

4 ~ f o r k = - - 1, l = O, f l - , ~ l ~ 4 I / ' ~ l / Io ~ + 1); 1

5 ~ f o r k = - - t , l > 0 , f ] - t < ~ T l o g [ ( [ f ] ~ ( l + l ) ) ~ l ( ~ + t ) + l ] ;

1 1))(h+t)/,t+ 0 6 ~ f o r k > 0 , I f lk < -k~_ - l - ( ! f I , ( /+

(b) I f f c o r r e s p o n d s t o q~ i n P t h e n a l l t h e a b o v e

f o r m u l a s 1~ ~ a r e v a l i d i f o n e r e p l a c e s I f{k , I f It b y

Proof . Consider q~EP" (the case of r is similar). 1 ~ is obvious

since max ]fl-2 is attained for f = Z + . In 20- -6 ~ our problem is to find

the maximum value of ]f!~ for a fixed value of ]fit. I f we take the quotient

ON EXPONENTIAL REPRESENTATIONS.. 367

of the weight functions in the l-th and k-th moments, we notice that in all

cases except 4* it has a unique minimum at ~.-~ O, whereas in case 4 ~ it

has a unique minimum for positive ~, at ~.= 1. It is obvious that the

maximizing distribution of density fdx should concentrate as much as

possible around these minimums. Since 0 ~ f ( x ) ~ 1 it is clear that f

should be a characteristic function of an interval. In all cases except 4 ~

f = X o , , ; t is calculated from IZo.tlt=ifl, and put into maxlflk=lXo,,Ik.

In case 4 ~, f = x a , b , where O ~ a < l q b ; from = iflo IZ,.bl0= b- -a , we get b = a N If]0 and put it in IX~.bl-t for which we calculate the

maximum for 0 < a < 1.

All the formulas in (a) are obtained in this way. The assertion in (b)

follows from the fact that the maximizing f on the whole axis must be

symmetric with respect to the origin, since all the weight functions are

symmetric.

The lemmas which follow will settle our main problem in this section

for functions in P', Q', P", and Q". We deal directly with functions in

P" and Q"; by correspondence (3.5), P ' and Q' are transformed onto Q*

and P" and the absolute moments of gt as well as of f are left unchanged

p x q~ be in in c a s e (b), T h e o r e m A,

by the correspondence.

L e m m a lO. L e t

w i t h ~ o o = l :

(4.9)

GO GO

q ~ ( O = 1 + ~ f ~ - -~ -~- e x p J " f dx

0 0

(a) T h e f o l l o w i n g r e l a t i o n s h o l d :

(4.10) 1 + Ilxj-1 = elfl-, cosjf[_2, ilxi_2 = eIJ'I-tsinIfl_2 ,

(4.10') tg tfl-2 1 + r Iz'

(b) I f IOj, or Ill* is f i n i t e f o r an i n t e g e r k ~ O , t h e n

1 0)*+, 0*+q, I i l, (4 .11) i~tl~ _< ifik = < ~ [ ( i l x l o + - - 19= ~ I-~-o ] "

F o r k = o t h e t w o b o u n d s c o i n c i d e , w h e r e a s f o r k > o t h e

l o w e r b o u n d is n e v e r a t t a i n e d ( e x c e p t f o r la~O) w h i l e t h e

u p p e r b o u n d i s a t t a i n e d f o r s p e c i a l f u n c t i o n s q0.


i i (c) I f tV!0 is f i n i t e t h e n f o r C<O

(4 .~z ) 1 < ~p(O < 1 + l/l___~0

Proof. (a) The relations (4.10) and (4.11) are obtained by putting

= i in (4.9) and separating real and imaginary parts.

(b) We apply the asymptotic expansion (4.6) to the two representations

of tV and write (4.9) in the form (4.7), where a t = I~1~-,, b, = - i / l , - , Since the polynomial qt has all coefficients positive, the first inequality in

(4.11) follows immediately from the second relation in (4.8). To obtain

the second we note first that by HOlder's inequality, for 0 ~ j ~ k,

hence

i l l . ~ I~'1. + q.(,,o ,,~,) with ,,; = , . i , - c ; - , ~ , , , J - . /~

But these values are the moments i~t~ of the mass l~t[0 concentrated at

the point 0----- ~ 0 ] " By the second relation in (4.8) the upper bound

is equal to the corresponding moment !f0]~. Since ep~ = 1 + ~9-----~. ~ ,

f o = 7~o,~+~uo, hence the second inequality in (4.11).

(c) An inspection of the first representation in (4.9) gives (4.12) with

i~,o instead of Ifi0. The replacement is justified by the equality (4.11) for

k = 0 .

Lemrna 11. L e t ~p be in Q" in c a s e (b) o f T h e o r e m A

w i t h ~oo = 1.

O0 O0

�9 dv f gdx (4.13) ,p(.~)-----l+.~ ~ . _ C - - e x p , . x-- r ,

0 0

d r < o , --1 <g(x)~o.

(a) T h e f o l l o w i n g r e l a t i o n s h o l d :

(4.14) 1 --iWr--* = e--e:--* cos :g;--2>O, iV!_2 : e--I~--t sin Igi--=,

Vi--2 (4.14') tg ,gt-~ = 1--IU5, ' e - ~ - , = (1 - ,~v i_ , ) 2 + Iv!L2.

(b) I f !v[k o r Iglk is f i n i t e f o r an i n t e g e r k ~ 0 , t h e n

(4.15) ig:k ~ ivik ~ (k + 1)lgik �9

(c) I f ,v,0 is f i n i t e , t h e n f o r ~ '<0


1 (4.16) 1 >__ ~ ( r ) >

1 Jr ]glo -E Proof. We proceed as in the preceding lemma. The inequality in

(4.14) comes from ]g[-z < arctgi//e ~lg!-' -- 1 < T~ (by Lemma 9(a) 2~

For the part (b) we use (4.7) and (4.8) with a # = - - Y t i - t , bt = [g]l-t.

The first relation in (4.8) is used. It gives immediately the first inequality

in (4.15). To obtain the second inequality we use the evaluations 1

Igls < ~ ( I g l ~ @ - i - 1 ) ~ s + , l t c ~ + , > )

(Lemma 9(a) 6 ~ ) which are attained for gO=--%o.t with

t = (tglk (k + 1))'/(k+'). t

The corresponding function ,po(~-) = 1 t--~" and lv~ = !glk(k + 1)

gives the second estimate in (4.15).

Finally to obtain (4~16) consider t P ( O- ' . From the exponential

representation of q~ it follows that ~p-t is in the case of Lemma I0 (c)

with a corresponding f ==--g. Taking reciprocals in (4.12) gives (4.16).

We are now ready to proceed with evaluations of moments for general

functions in P (or Q). We shall consider each case of Theorem A separately

and will use the decompositions introduced in the first part of the proof

of Theorem A. Lemmas 10 and 11 applied to the factors # ' and *" will

allow the evaluation of moments on each half-axis separately. We refer for

notations to the proof of Theorem A.

C a s e (a). q~ is normalized by assuming ~ = 1.

I) We assume only f i x - ~ @(1,1) . We write

4al, ' ' ; I 1 I - ~ q g = l + - ~ r - - + - ~ - ~.__~ ~2~.. 1 d~(X)=exp ~ .

By putting : = i and using O r - X - - ) - 2 = Ift-~- = ~ , we get

~3~---- e l f -x- I - i cos !fi-2, 1 + i~i--: = el l-z- l-1 sin If t-2, (4.a1') ~i

c o t g I / I -~ - 1 + 1~I-= ~ , , ~ - z - ~ - , = 13' + (~ + !~I_~) ~-

The only quantities to compare are I f - x- i - , and I f - X-i-2 on the

one hand and Ilxl_2 and t~i[ on the other. This purpose is achieved by (4.al')

together with T if -- % [-2 ~ arctgV e i i - z - I - t -- 1 (Lemma 9 (b) 2~


I1) We assume f - - X - ~ @ ( 2 , 2 ) , hence F e ~ ( l , 1).

(4.a2) ~-q~ = l + -7;-- + =: exp - - - - d x = ~p" �9 I p " ~ ~ ~ - ~

If ! f - - z - i k < oo for k ~ o , then IV[a-t < oo and applying the

developments (4.6) and (4.7) to (4.a2) with at =- - [3o0 , a t = ~tj-2 for

2 ~ j ~ k + l , bt =- - - ( f - - z-)1_t for l < _ l ~ k + l , we obtain from (4.8)

a series of equalities, the simplest ones (by (4.8') and (4.8")) are

6= = - - ( f -- X-)0, [F[o = If - - X-I1 - - T ( f - - X-)2o , (4.a2')

1 1

i f - - X-t1 = !~t!0 + -2- ~ , ( f - - X-)2 = FI - - ~oo [bt[o - - ~- ~ .

These equalities lead to evaluations for i~ool and IF[I for even 1 or

for i f - X-it for odd l. To obtain evaluations for other absolute moments

and for mixed moments we use the relations

(4.a3) d F - ( k ) = k , " (k)dv'(~) , dF+(k ) = kV ' (k )dv"(k) .

We notice that q~' and ~p" are in the setting of Lemma 10 (~p' by

correspondence (3.5)). Hence, by (4.12), for l ~ O and l-------2

i~' i,+, _ iF-I, <= iv' Ii+, + If'j0 fr (4 a3') <= tF+t, --- tv';,+, + If' - z - J 0 iv%.

For l = - - 1 we get

; t q ' # t

tv ~o - iv" I- , ~ iF-I- , ~_ I"' io - i," I-, + tf to i" 1-, ] r ~ 1 n - - n

~ Io - !~ !-~ <= IF+I-, <~ Iv':o - I~ t-~ + If' - z Io Iv !-1. The second equality in (4.10~, Lemma 9 ( a ) 2 ~ and the last equality

in (4.a1') (where we put ~i = ~ + F - t ) give

iv']_2~_A, iv"[_2<A with A---#(Ifl| (4.a3") Iv' f ' __ --' _ - o = . - Z-lo < I~ J-, + A !~ 1o = !f"io < !F+i-, + A .

We are now able to obtain all the desired evaluations by using

inequalities (4.11). The simplest are as follows.

(4.a4) if - - X-i0 --~ IFI-~ + 2A, I~tl_l ~ [f - x-10 + ~-if - x-]~; ~ ' 1 t 1 J r '

in the last one we used (4.a3") and !v l-a ~ y Iv!0, iv"]-I ~ T I ~t ]o,

1 A 2 1 = (,FI-1 + A)', IFIo < I f - z - I , + 5 - I / - Z - ] ~ . (4.a4') !f-x-l,<__lF0+ ~__ +~_l~

C a s e (b). q~ is normalized by assuming ~oo = 1.

I) We suppose only f ~ @ ( 1 , 1 ) and write


(4.bl) q~ = 1 + -j" dix(~.) _ exp PJ fax z - ~ x - ~ " = ~P'~P"'

[IX[_= = e/-* sin If i-2, I q- Ix-, = e/--a cos [f]-z

(4.bl ') tg [f[-~ -- 1 [~tl-2q- Ix-, ' e2/-* = [Ix[2-2 q- (1 + Ix_,)2.

I n t h e p r e s e n t c a s e i t i s n o t p o s s i b l e i n g e n e r a l t o

e v a l u a t e t h e a b s o l u t e m o m e n t ] f l - ' by t h e m o m e n t s ]Ix[-2 t

a n d ]IxI-1 �9 A counter-example is provided by the function q0 (~) == 1 q- -t------~ '

t > 0 . Here

t t 2 1 r, Ixt-2 = t2q_l , iIxj-i = P + I and t f l - I = '~- log( t2 "q-!) '

It is possible, however, to evaluate IIxt-~ and iIxI-1 by VJ-1. A trivial

evaluation IIx[_2<elJJ-, comes from the first equality in (4.bl '). Another

one, better for small ]fl-*, comes by using in this equality the evaluation

of Lemma 9 (b) 2 ~ For [Ix]-I the equalities in (4.bl ') do not give directly

a result. To obtain it we use the function 0o

dIx 9 + = i + , Z-~ ' "

0

We notice that qo(~')--q~+(~')<0 for r > 0 , hence by the comparison

theorem f"(x)>=f+(x) and thus rf+i_,<If"!_~. Writing IIxl-' = 2!Ix+!-'--IX -~,

using the equalities

l ~ t+ l - , - : e ; / + I - ' cos i f + t _ 2 - - 1 , gL-1 = e / : - I cos l f t - -2

and then applying Lemma 9, 2 ~ (a) and (b), one arrives by a careful evaluation

to a bound 3 ( e ! I i - , - 1). Our evaluations can be written:

(4.b1") [Ix]-2 <= min[d/ l -1 , 2 e ( I t 2 ) i I l - l l / e l 1 1 - 1 - - 1], IIx]-a =< 3 ( e , f , - i - - 1).

II) We now assume f ~ @ ( 2 , 2 ) , hence I f ]0< ~ , iPI0< c~. If

I f l k < ~ for k ~ 0 , then also IIx[k<~ and we can apply developments

(4.6) and (4.7) to (4.bl) with ai = Ixi_~, bi = - - f j _ , and obtain from (4.8)

a series of equalities the simplest of which are

(4.b2) !Ixl0 = Ifl0 (Verblunsky's theorem) 1 1

I x ' = f , - T l f l ~ , [Ix] ~ = ] f l 2 - [ f l 0 f * - b ~ l f l ~ , etc. (4.b2')

1 2 1 3 f ' = P" q- T !Ix]o, If!2 = !IXl2 q- 1~!o Ix, q- 3-]~tlo , etc..


We obtain here evaluations for absolute moments of even order of f

as well as of I~. For moments of odd order and for mixed moments we

have to use the relations

(4.b3) d V- (4) = 't'" (~) dr ' (~), dV+ (4) = , ' (~) dr" (4).

The function q~" is treated by Lemma 10, whereas lp" belongs by

correspondence (3.5) to Lemma 11. By inequalities (4.12) and (4 .16) we

deduce from (4.b3) for l ~ 1

(4.b3)' Iv" l, ~_ It*-[, _~ iv' l, + lf'lo Iv' i,-, iV+it ~ {v" I' --~ IV+it + if ' l0 l~+l,-t �9

The left inequalities are valid for l = o also. These inequalities,

together with (4.11) for ~p" and (4.15) for ~p' lead to all desired evaluations.

We have

(4.b3") J'!0 ~ [~t-[0 ~ ifi0, iV+i0 --<_ if",0 ~ iVl0,

I/'i,~_i~-I,~_(l+x)~/'l,+lff'!oll'I,-,, Iv+[,<i/'l,, for I ~ I .

An upper bound for If"l~ is more complicated. For l = 1 it is

(4.b3") !f"!, -<-- IV+I, + P~'I~ + iV-!o IV+lo �9

C a s e (c). q~ normalized by ~oo = -- 1. This case is completely parallel

to case (b). All formulas and evaluations of case (b) transfer to the present

case by reflection : absolute moments of f , f ' , and f " are replaced by those

of l - - f , 1 - - i f , and l - - f " respectively; absolute moments of I~, ~ - , and

V + are replaced by those of V, V+, and V-. However, in the present case

if there is a mass at the origin it should be assigned to ~t- (to V +, in

case q~E Q), in order to have our standard convention satisfied for the

reflected function.

C a s e (d). q0 normalized by: total mass V6= [Vio = 1.

I) Assume only f -- X +E @ ( 1 , 1 ) . We write

(4.dl) --~'q~ ( 0 ----- - - : f ~ - - ~ ----- 1 -- ; ~'dla (~ ' )--exp : f - z + dx : ~p' xp ~

~t (4.dl ' ) ( f - -x+)-2 = Ift-2 -- -~-, tVi-2 = e - l : -x+ l -a sin [f l-2,

~ , _ , = ~-~/-x+~_~ cos l/i-~, tg )ft-~ = tt~l-,,

e- , i / -x+t_, = IvIL, + p-L,-

ON EXPONENTIAL REPRESENTATIONS... ~7~,

In view of our normalization we have here obvious evaluations for

i~t-z and I~tJ_:. Also the last equality gives an evaluation for I f - X+I - t .

We summarize it in

(4.dx') 1~,1-~ --- 1, tlXl-, ~ V ilx!-2 (1 - I~1 -~ ) , If - x+t - ' ~ - l o g 1~,1-~ �9

To obtain evaluations for moments of f " and f # - - X + we introduce

the functions 0 QO f+- /-++

(4.d2) cp-- (~') = ~_ r , cp+ ( 0 = , ~_.~. �9 - - 0 0 0

These functions are also in case (d). For the corresponding f - and

f+ we have f - ( x ) = l for x>O, f + ( x ) = O for x<O. We write 0 0

(4.d2')

( - ~ , r ( O = Ir~-~o 1 - - -

-r ( 0 = l~+]o ( : \

!~-1o k--~ d x - r ' - - 0 0 - - 0 0

~0 00 : ) --+ z ~a~+ (t) = i~+l ~ exp ~ J.~-~--~[ dx. IF+t0 ~- -~ v ~ - . .

0 0

Since q 0 - - q 0 - > 0 for 7 < 0 and ~p--q0 + < 0 for ~ > 0 , we obtain by

the comparison theorem

(4.d2") f ' (x) <__ f - (x) - - Z+ (x) , Z+ (x) - f" (x) < z+ (x) - / + (x) .

The functions qg-(~) and ~--~IT g~ (~) are of the form (4.dl).

By the last equality in (4.d1') we get

i~-Io If 'f-, -<-!f- - z + i - , = l~ + tvt-IL, '

(4.d2"') l~t+]o

+f" -- z+!_, <= log V ! . + G + !,~,+lk,

II) We assume now f - - z + r @ ( 2 , 2 ) . Again we apply (4.6), (4.7),

and (4.8) with a i = --~ti, bl = - - ( f - - z + ) t - t and obtain a series of equalities

(4.d3) p t - - - - ( f - - x + ) o , l~t12= If - - x+It 4- -ti ( f -- x+)~ , 1

~t3 = -- ( f -- X+)2 -- (1 --X+)o tf -- Z+t, -- ; ( f --Z+)~ etc..

We use now the formulas

~_ Z , , (~ ) (--dr" ( t ) ) . a~,- 0') = - - * " O')' tv' (X), at, + (Z) = W

Since ap' and , " are in the case of Lemma 11 (the former by correspondence


(3.5)) we can apply the first inequality in (4.16) to obtain

- - 4 d ~ - (4) _ dr ' (4), 4dIx+ (4) __ - - dr" (4) ,

and hence by (4.15) for l > 0

IFI,+, -<_ Iv'l, <= (z + 1)tf'!t, I~x+It+, < tv"l, <= (l + 1)tf" --X+l' �9

On the other hand since the functions (4.d2') are also in the case of

Lemma 11, by using (4.d2") and (4.15) we get

1 z t,= t~+10 IF I,+ [f,[,<lf-_x+],<l_~_~olvt-i,+, ' ! f - _ z + l , < = l f + _ + < _ _ _ L + ,

Summarizing, we write

(4.d4) I~-Io If'l, < IIx-I,+, < (Z + 1)If'i,, t ~ o , t~+[0[f" -z+[ , =<_ IF+l,+, <= ( t+ 1)[f*--X + j,, l ~ 0 .

These inequalities lead to evaluations for all mixed moments of

f - - x + and ix.

5. Behavior of Ix and f at a finite real point. /: lpplications.

All our previous developments were aimed at the analysis o f the

behavior of a function q~ in P at ~ and the behavior of the corresponding

measure Ix and function f at co, described by the finiteness of the moments

tl~I~.k and Iflk,k.

To analyze the behavior at a finite real point a we will use the

transformation ~.r* = ____1 which transforms a into ~ , oo into O, a + i

into i, and the upper half-plane onto itself. The inverse mapping is

1 ~ " - - - - - a - - ~ - and tp (~') is transformed into

(5.1) ~ * ( r ' ) = ~ a - T ~ - , ~ ( . r ) = ~" ~ .

All our previous results will be considered as statements about c0 ~ (~~

and translated into statements about q9 (~'). These new statements will be

referred to by naming the previous corresponding statements and adding

"rel. a", e.g . Theorem A rel. a, Lemma 3 rel. a, etc.

In order to make these translations easily and consistently we introduce

the following conventions and definitions.

C o n v e n t i o n I. In general, measures will include masses at eo ;

with our previous notations e will be considered as Ix [ ~ ] . I f we decompose


a measure into its restrictions to the half-lines [ - - ~ ; $] and [$ ; + o o ] ,

rt = ~t~-+ ~t~, the mass at oo is included in ~t~-, the mass at ~: is in ~ ' .

With this convention the formula c~ = a ' ~ o of Theorem I is included

in the formula

art (L) = r (~) d~" (~) + ~" (;,) a r c (~).

C o n v e n t i o n II. For Stieltjes integrals we will write

meaning that

d d < d < d

j " , . ( , , f ' f , c c < c c <

the masses at both endpoints are included or mass at c

excluded, or mass at d excluded or both masses at e and d excluded. The

mass at infinity will always be excluded in integrals. In an integral over

the whole real axis, if we want to exclude the mass at c, we will write

< c < < c +c r

f = f + ] , -co c <

This convention does not change the meaning of all integrals in the

previous sections.

With these conventions the representations of r (C*) rel. i and tel.

will now be written

< o <

r r ,~'t~] c. + 0: + ,._=.t0] + f [~. ~-z.- z.~+t~" ]acct.),

< o <

rt ' [o] f (5.2') q,'(,') = ~" [~1 c" + 13~, + ~ + a~t*(x') L . _ _ ~ - - �9

The representations o f q9 (~') rel. a + i and rel. a are < a <

a~ (~). (5.3) ~ ( C ) = r t [ ~ ] ( ~ - a ) + ~ , + , + a - c ~-~ ( X - - a ~ t

< a <

(5.3') ,~ (0 = ~ [~1 ( ~ - a) + ~ + ~ + ~-C ]dg(t ) . J~--a

Defini t ion I. P ' P" a n d a a r e c l a s s e s o f f u n c t i o n s ( p ~ P ,

s u c h t h a t f o r t h e c o r r e s p o n d i n g f , f ( x )=O f o r x>a o r


x < a r e s p e c t i v e l y � 9 F o r q ~ P w e d e n o t e :

f ' a (x) = f (x) X- (x - - a ) , f~ (x) = f (x) X + (x - - a ) .

Defini t ion II. T h e m o m e n t s r e l . a a r e f o r m e d as f o l l o w s < a <

[~t!a:_.7. = I~ [~ ] + (~.--a) 2 -a t- 1 '

< a <

for - - 2 < k < o , i~1.;~ =

< a < ~ a~(x)

!<.,o ~[o~]+ i~_a!, ; +

I~la,..' = 1~7f~ + t ~ l a , ~ .

I~.-a:+' [ ( ~ - a ) ' + ~] '

< a <

for k > O , i~!o,~= ;

< a < d~t (x)

~'a,o---- I~'ta,o;

for an integer k > O,

< a <

f d~ (X)

The moments o f f rel. a are defined similarly with the measure f dx .

The classes 9Y/a(h,k), ~ - ( h ) , and ~rJ~a+(k) are defined as the

classes of measures ~* with supports on the whole real axis or on ~. _~ a or

on ~ . ~ a respectively for which the corresponding moments I~tt~,~.k, [lxl~,h

or I~t!a,k are finite. The classes @~(h ,k ) , @~'(h) , and @+(k) are

defined similarly.

The results o f the previous sections can be translated into statements

rel. u by using the following

L i s t o f C o r r e s p o n d e n c e s : (1) ,0 (O = ~* (~') = �9 ~ , d~ (~) = ~ ~ * (z') = ( ~ - a y d~t"

~ 7 ( ~ . ) = '- . . ~ , . ~. �9 .-~.~-~--d~+(~.), d~O. ) = . - p T a ~ - O . ) , ~ [ ~ ] = [o] ,

~ [ a ] = ~ ' [ o ~ ] / ( x ) = f ' ( : ) = f ' ( 1 ) , ~ , f ' ~ ( x ) = f * " ( x * ) ,

/~ (~) = / * ' (x ' ) ,

x.- (x) = z - ( x - a ) = z + ( : ) , z. + (x) = z+ ( x - a ) = z - (x ' ) ,

= * = * [ f t a , ~ , ~ = l f* l~ ,h , f a t k *


Classes P'~ and P~ correspond to P" and P" respectively.

Classes ~ a ( h , k ) , ~ - (h), ~t~ + (k) correspond to ~ft (k, h), ~0A+ (h)

and ~-~-(k) respectively. Similarly for classes @.

Representations tel. i for q~*(~") correspond to representations

rel. (a+i)for q~(f); representations rel. oo for q0*(f*) correspond to those

rel. a for q~ (~').

As examples of application of the "relativized" statements we give

the following theorems.

Theorem IV. For q ~ P , ~[a]>O if and only if if--x,~i~,-~< ~ .

W e h a v e a l w a y s Ix[al=exp(oa+,--tf--x~lat_,). The first assertion is Theorem A (a) rel. a. The second comes from

w 1, II) since, if {f--x2t , , - t< oo,

(a-- r , ep (~') ---- exp {o.+, + f [ - x--:l (x--a) 2(x-a'+ l] ( f_z+,dx}

= exp oa+~ --If--x~i~--t + x--r x--a

Theorem V. A f u n c t i o n q~(.~) is r e p r e s e n t a b l e in t h e

f o r m <a<

!~t!--2+t~la,-,<e~ if and o n l y if it is r e p r e s e n t - w i t h d ~ O ,

a b l e by

exp x-- f x--a

w i t h 0 ~ f ( x ) _ ~ l and ! f l a , - , < ~ . T h e e q u a l i t y 18i~,0=[ft~,0 a l w a y s h o l d s .

This is just Theorem A, (b) with ]300 = 1 translated rel. a. The

equality corresponds to Verblunsky's equality (4.b2). Theorem VI. A f u n c t i o n V(.f) is r e p r e s e n t a b l e by

f ~dv(~) 1 - - , ~_~, w i t h d v ~ o , !vl0~l a n d v [ O ] = v [ ~ o ] = O i f a n d

f g(x)ax o n l y i f it is r e p r e s e n t a b l e by exp x--~" w i t h t g ( x ) t ~ l ,

xg(x)~O and ! g t - l < ~ . T h e e q u a l i t y 1--!vl0=exptgi0,- t a l w a y s

h o l d s .

3?8 N. ARONSZAJN and W. F. DONOGHUE, Jr.

W e consider the function

,~ (.c) = _ - - - ( , 6,) - - ~ + ,~( ~._;

which is in P in case (d) o f Theorem A with measure d~t = d~ except for

~t[O] = 1 - Ivi0. q0 is normalized as in case ( d ) w and if f is the

corresponding function then g= f - - X + satisfies all requirements of the

theorem. The argument is obviously reversible. The last equality in the

theorem follows f rom Theorem IV applied to q~ with a = O,

T h e o r e m VII . A f u n c t i o n LO(,C) i s r e p r e s e n t a b l e b y

l+f ~.dv(~,) w i t h dv20, ]vI0<oo a n d v [ O ] = v [ o o ] = O i f a n d ~ , _ r

o n l y i f i t i s r e p r e s e n t a b l e b y e x p f g(x) dx

z w i t h ig(x)j ~ 1 x - -~ -- '

xg(x)~O, [g[_t<oo a n d Iglo,-,<oo. T h e e q u a l i t y

f g(,O,r exp - = exp (IgJ-'~ + Iglo,-1) = 1 + JVio r ,

a l w a y s h o l d s .

W e consider here the function

1 1 ] ~.2 dv ~,--~" ~. J "

q~ is in P and is in case (a) of Theorem A and corresponds to dfx = ~2dv,

t~[oo] = 1, and a function f with I f - - x - I - l < o o . Obviously g = f - - z - , hence Ig[--i < oo . On the other hand q~ is obviously in case (d) of

Theorem A rel. 0 with h,10,o= x + Iv]0; hence t g ] 0 , - t - - I f - x - 1 0 , - 1 < oo.

The last equality in the theorem is obtained by taking ~"->-0 in an angle

in the two expressions of ~p (~').

R e m a r k . The two last theorems were the ones which instigated the

f research presented in this paper. The expressions ~(~ ' ) = 1-T- ~._~.

appear as determinants of one-dimensional perturbations of self-adjoint

spectral problems (theory to be developed in a forthcoming paper by the

authors). The behavior of the exponential representation sheds light on the

change in spectrum caused by the perturbation. A special case of Theorem VI

was considered in [2']. (12) W e notice that equalities and inequalities o f w 4,

12. In this paper, in Lemma 3, the function ~ =- ( ! /g )q was considered rather then q, in the case of a point measure v distributed on a sequence of points converging to 0.


especially cases (a) and (d) give a series of relations between moments

rel. ~ and rel. 0 of v and g.

As another example we now apply our results to the study of Weyl's

spectral measures corresponding to a Sturm-Liouville problem in the limit-

point case. If the boundary condition at the regular end-point is given by

the usual parameter rg, - - - ~ - < ~ - ~ - , then the l i m i t p o i n t me(~')

~ 9 ~ ,

m~ (O = mo (O cotg (~-,9) - 1

mo (~') + cotg (~-- 19)

This relation can be written (see [2])

--1 (5.4) [too (~') -- cotg (0--T] [mr (~) -- cotg (~--~)] -- sin2 (z--~) "

The measure p# corresponding to mo(O is the W e y l ' s s p e c t r a l

measure corresponding to O. The values ~ for which ~#[k]>O are the

eigenvalues corresponding to ~.

Put wo,.r(~)= mo(~)--cotg(0--T) and let ]o,r be the corresponding

function f . (5.4) gives immediately, for T=;e ,9,

--1 (5.4') wo,r(~)wr,#(O= sin2($_T ) , fo,r(x)+f~,o(x)= 1.

The behavior of the measures [tr at o0 is very well known (see

Gelfand-Levitan [7], Krein [10]) especially when the equation can be trans-

formed into the usual form --x'+ (q--~)x = 0, which is feasible under

slight regularity restrictions on the coefficients. These known facts are

summarized in

S - -LI ) b t o [ o o ] = 0 f o r a l l {~; ~t~'E~lJ~-(k) f o r a l l k;

~g �9 u + Ckl "~ ~s/ ' f o r k ->- ~ o ~(~)~112 f o r k-->- o~ a n d 1,9]< T , ,-~/.~, ,

We shall investigate the behavior of m#(~') and Po at a fixed real

point a. Our results will be valid also at oo but then they will confirm

only in somehow weakened form the above statement.

S--LID I f f o r s o m e b t, ~%[a]>o a n d b%E~a(h,k), h~o, k>o, t h e n f o r a l l *=J:$, b t r [ a ] = 0 , [ - t r E ~ / a ( h + 2 , k + 2 ) ,

is a function in P which for two different values of the parameter,

satisfies the well known relation

38o N. ARONSZAJN and W. F. DONOGHUE, Jr.

mr (.~) ----- cotg(x--~9) + k - r ~,--a" d~(~),

/ dw(x) _ 1 ;~!~176 - - I ~ - a l ~ ~o [~1 sin~ ( , - 0 )

C o n v e r s e l y , i f f o r s o m e ~, ~ t r [ a ] = 0 a n d V t r e ~ a ( h + 2 , k + 2 ) ,

h > - 0 , k > = 0 , t h e n t h e r e e x i s t s a u n i q u e ~9 w i t h ~ % [ a ] > 0

a n d ~ o r

In fact wo , r is in case (a), Theorem A rel. a,

by (5.43 . f , , , - - L + ~ @ ~ ( h + l , k + t ) ,

a

Wr.o is in case (d), Theorem A tel. a and ~ E ~ a ( h + 2 , k + 2 ) . The

equations follow immediately from the form of w : , o and from (5.4). The

converse follows by a reversed argument.

It should be noticed that a series of inequalities can be obtained

between the moments [~rta;i+2,t+2 and ]~,]a,,,i by using the "relativized"

inequalities o f w 4, especially cases (a) and (d).

S--LIII) I f f o r s o m e 3, bLr[a]-----0 a n d ~EgX~(1,1)--~(2,2) t h e n f o r a l l o t h e r v a l u e s o f r, e x c e p t o n e , t h e s a m e

r e l a t i o n s h o l d . I f ~ i s t h e e x c e p t i o n a l v a l u e , t h e n v o [ a ] - - 0

a n d ~ % E ~ a ( 0 , O ) - - g T ~ a ( 1 , 1 ) . F u r t h e r m o r e

mr(~) "--- co tg(T- -~) + ~ , - -~ ~ - - a dp~00

f o r a l l r # O . Oa)

In fact, for ~' :# T and except when cotg (z--T') = mr (a), wr. ,, is in

case (b) or (c) o f Theorem A rel. a, hence f r , , , or 1 - - f r , T , is in

@~ ( 1 , 1 ) - - @, (2 , 2), by (5.4") 1 - f , , ~ or f ~ , , , ~ @ ~ ( 1 , 1 ) - - @ ~ ( 2 , 2 )

and wr, ,r is in case (c) or (b) respectively.

Notice that for a-----oo our last statement holds with ~ : = - - . 2

By using Theorem B one could add some more detailed information to the

last statement.

13 It seems difficult to characterize completely and directly the behavior of ~t O for the exceptional 8.. An indirect characterization would be to add to the conditions in the text the following: --t/too(g) has a measure in ~rf~(1,1). Notice that by (5.4), --l/too (0 •ffi too, (0 with 8., - - 8. ~ ~/2 mod ~.


6. /~bsolute continuity of ~.

Throughout this section q~(~') will be a function in P, F the associated

measure in ~0~(0,0) and f ( x ) the function in (~ (0 ,0 ) occuring in the

exponential representation of q0. We are interested in tl:e relations between

f (x) and F which serve to distinguish the absolutely continuous part of ~.

By a s u p p o r t of ~t we shall understand a set S whose complement

has p-measure zero; a support will be called m i n i m a l if any subset R

of S which is also a support of ~t differs from S by a set of Lebesgue

measure zero. A support of ~ is called t h e c l o s e d s u p p o r t if it is

the smallest among closed supports.

We write the Lebesgue-Jordan decomposition of F as follows:

~ = ~0 + Ft, where p0 is absolutely continuous with respect to Lebesgue

measure and ~t is singular; ~t: can be decomposed into V*= ~ ' + ~ where

~" is a point measure and ~' the singular continuous measure. A theorem

of de la Vail& Poussin permits us to assign supports to these measures

as follows.

~ ' = E [ ~ exists and 0 < ~ <

exists and equals + ~ 1

Sp = E [~ (~.) is discontinuous at ~, = ~]

f

s p = - s ; .

j n

We note that these supports are minimal and that S ~ ~ S~ = Sp r Sp = O, f S~ = Sp U Sp.

Another assignment of supports can be made with a definition which

does not explicitly involve ~z but rather the behavior of q~(~) near the

real axis ; we call the following system of supports the s t a n d a r d supports

(see [2]).

S~ exists and is finite and limImq~(~')>0 when ~'+~ in any

angle]. S l = E [Imq~(~')-~ + ~ when ~-~ ~ in any angle].

S" = E [Ira q~ (O + + o~ and (~--~) q0 (O "~ 0 when ~" -->- ~ in any angle].

S" ---- E [lira (~r_r) q) (~-) > 0 when r .~ ~ in any angle].


Here again we have S ~ = S ' / ~ S " = 0 and S t = S ' L / S " .

In view of (1.7') we see that S ~ , C S 1 and by w S 1 is o f Lebesgue

measure 0. Also f rom w 1, IX) and X), it follows that the supports S ~ and

S o differ only by a set of Lebesgue measure O, while from w 1, I I ) it follows

that the enumerable set S~ coincides with S ' . Accordingly the standard

supports are minimal. It should be observed that the definition o f the

standard support for the singular measure tt * is open to the objection that

it may happen that S * is not empty even when !~ is absolutely continuous.

Hence a better support for ~0 can be defined as the intersection of S ~ with

the closed support o f I~ J, the latter set o f course being the complement of

the largest open set upon which ~t is absolutely continuous. Such a set is

an enumerable sum o f open intervals upon which [L is absolutely continuous,

and we recall that ~* is absolutely continuous on an open interval a<3.<b if and only if ~t is absolutely continuous on every closed subinterval o f that

interval. W e do not give a direct definition of the closed support o f ~t 1

and will give instead only sufficient conditions for the absolute continuity

of ~ in an open interval.

On the interval a<),<b let the measure ~t~ correspond to the density 1

- - I m q~ (3.+i~/); it is evident that Pr, is absolutely continuous and that the b

total mass f is uniformly bounded as ~ converges to 0 (see w t, IV)).

Since the functions q~('~+zti) converge in P to q~(~') as ~--~O, the measures

p~ converge to ~t as described in Section 2. Thus ~t will be absolutely

continuous on any closed subinterval [a ' ; b'] if and only if the system tt:~

is uniformly absolutely continuous on such a subinterval, or, equivalently,

if and only if the densities p'~ (3.) converge weakly in L 1 ( a ' < 3 . t and this, in turn, is a consequence

of the LP convergence of ~p (~+iz/) to q~ ($). By a localization of the theorem

of M. Riesz (9 1. XI I I ) we finally arrive at the following statement which

will be used later:

I n o r d e r t h a t p b e a b s o l u t e l y c o n t i n u o u s i n ( a ; b )

dtt a n d -d~- b e i n Lt'(a';b'), p > l , i n e v e r y c l o s e d s u b i n t e r v a l

[ a ' ; b ' ] , i t i s n e c e s s a r y a n d s u f f i c i e n t t h a t f o r e v e r y s u c h

O N EXPONENTIAL REPRESENTATIONS... 383

b '

i n t e r v a l fj~p(~+i~)fd$ be u n i f o r m l y b o u n d e d w h e n 7 + 0 . ~e

We next define supports of ~0, ~1, V', and ~t" in terms of the

function f .

Obviously S O is the set of ~'s where Log~p(~:) exists and is finite

and where 0 < I m l o g q ~ ( ~ ) < ~ . This set differs only by a set of Lebesgue

measure 0 from the set ~0 of all ~'s where the derivative of x

F (x) = f / (t) dt 0

exists and lies between 0 and 1 and where the singular integral

f dx (6.1) h->-olim l J" x--: + f x--: f dx--l'

: - t ~+h

exists and is finite.

As support ~1 we take the set of all ~'s where the singular integral

(6.1) exists and is + oo. In view of (1.8) and (1.8') (in the present case,

t x~(~7)[ = < • ~' is the set of ~'s where

lira Re Log q0 ($ + i72) = + oo. ~--~0

This set may be larger than S 1 but we will have

:~o n ~1 = S O t~ Sl = o .

We put S " = S " and ~ , : ~ 1 _ ~ - . Theorem IV allows us to

characterize S" as the set of all $'s such that

f - x~l~;_, = f f(x)a~ + i (:-f(~))a~ (:--X) [(~:--X) z -+- 1] ,J'" 1]

< 0 o

' ( x - ~ ) [ ( ~ - x ) ~ +

Obviously for such ~ the singular integral (6.1) is + ~ , hence again

~O=~'u~" and g ' ~ " = o . We now turn to sufficient conditions in terms of f assuring the

absolute continuity of ~t in some open interval ( a ; b ) . Our conditions will

actually give more, namely that the density dI.t/d~, is in Lt'(a';b') for any

closed subinterval [a' ; b'] and for some p2>1.

In order to describe conveniently our conditions we have to define

the function f with more precision; till now it was any function in its

equivalence class,


We define the domain D of f (= the Lebesgue set of f ) as the set

of all x's for which the indefinite integral F(x) of f has a derivative. iF (x)

For x ~ D we put f ( x ) = dx From now on whenever we write f (x)

it will mean that x E D and f (x) is the value just defined.

L e m m a 12. L e t o ~ = A ~ = f ( x ) ~ B ~ l f o r - -o~<a<x<b<o~

a n d l e t B - - A < 1; t h e n t h e m e a s u r e ~t is a b s o l u t e l y

c o n t i n u o u s in (a ;b) w i t h a d e n s i t y in LP(a';b') f o r a n y

1 c l o s e d s u b i n t e r v a l [a ' ;b '] a n d a n y p < B------A-"

Proof. By the theorem of M. Riesz it is enough to prove that b'

f Iq: ($ +iTl)iP d$ is uniformly bounded as ~l ->" 0. Decompose f = f t + f2 af

with f , = fx~,b and take a corresponding decomposition q: --- q~l q~2 which

is well determined except for a multiplicative positive constant. Since q~2 (~')

is regular and positive for a<~ 0 p ( a - - c ) = *[I+p(B--A)]<i (6.2) p (A--C) = -f , y .

Consider now the function

, , ( : ) = , �9

t \ ~ - a / 1

It is immediate ly verified that V has an exponential representation

with a function

(6.4) g (x) = p [f , (x) - - Cxa,b (x)] , b'

hence, by (6.2), V is in P and ./ 'ImV(~+i~)d~ is uniformly bounded. a l

Since

" ' i i,p, (~+i~) ip sin [Im Log V (~+i~)] Imq~(r = i ~ + i ~ _ a i

and ImLog~p is, by (6.4) and (6.2), uniformly bounded away from 0 and bf

for ~" near [a ' ; b'], the uniform boundedness of .f Im V (~+i~?)d~ implies b ~ af

that of f!Tl(,~+i~?)iPd~, which finishes the proof. r

Consider now a finite open interval ( a ;b ) . We define


(6.5) m ( a , b ) = . } ( f ; a, b) -~ sup[f (x")- - f(x')] f o r a l l x ' ,x" s a t i s f y i n g a < x ' < x " < b .

This number can be called the u p p e r r i g h t o s c i l l a t i o n of f

on (a ; b). Obviously 0 < (o (a , b) < 1 .

Lemma 13. F o r an o p e n i n t e r v a l

t h e n ~t is a b s o l u t e l y c o n t i n u o u s i n

LP(a ' ,b ' ) f o r e a c h c l o s e d s u b i n t e r v a l

2 p < 1 + o ( a , b ) "

( a ; b ) l e t 0 } ( a , b ) < l ;

dp (a ;b) a n d d~. i s i n

[ a ' ; b ' ] a n d f o r e a c h

Proof. We will write { o : m ( a , b). Consider the subsets 1>1 and P2

of (a ; b):

{E(a ;b} la<x<~ z I {~(a;b) t { < x __ - - - or 1 + { o 2

f ( x ) ~ . . . . . and our lemma would follow from Lemma 12. Hence, we 2

may assume that P t ~ - O ~ P 2 . (6.5) implies that if x t E P l and xaEP=

then x l > x z . Consequently there exists a point c separating Pt from P2.

It is clear that

f ( x ) ~ . . . . . . 1--m for a < x < c , f(x)=<----l+m for c < x < b . 2 2

The function

( : - -c l-('-w)12

,p (~} = l . . - ? -TI ,p (;') is then in t ' with a function g in its exponential representation given by

1 - - o} 1 -I" {0 g (x) = f (x) . . . . . 2 - Z-,~ (x) and satisfying 0 :<. g (x} :< . . . . 2 for a < x < b.

b'

Thus, by Lemma 12, j" I'q (~'+i72) J'd~ is uniformly bounded the same as tP

b* b*

~i(p(~+i*l),t'd-~ -- { ap (~+'i~1) IP r +*~2 - c 'p(1-w)t2 �9 - Y q - - i , T - a L e e .

a I c~*

The function {,) (a , b) is a decreasing function of the interval (a ; b),

hence we can define for any real x


(6.6) ,,~ (x) ~ t0( f ; x) =-= lira o ( f ; a , b). a4x,O~'x

Obviously 0 ~ t0 (x )~_ 1. Standard arguments show that tt~ (x) is upper

semi-continuous, that on each closed set it attains its maximum and that

the set where o) (x)< I is open.

T h e o r e m VIII. T h e m e a s u r e ,u is a b s o l u t e l y c o n t i n u o u s

o n e v e r y s u b i n t e r v a l o f t h e o p e n s e t G = E [ o a ( f ; x ) < l ] . dlx

O n e a c h c l o s e d s u b i n t e r v a l [ a ; b ] o f G t h e d e n s i t y - ~ i s

2 in LP(a;b) f o r a n y p <

1 + max ( ~ ( x f " xe[a ; Ol

Proof. It is enough to prove the second assertion. Put m =-~ max o (x), a,~x<__b

2 then 0 ~ m < l , m < - ~ - - - 1 and for each x ~ [ a ; b ] there exists, by (6.6)

an interval (a, ;b , ) containing x such that

I '- 1 m(ax, bx)< min 1 , - ~ - - - t .

We can cover [ a ; b ] by a finite number of such intervals ( a , ; b~ ) and

therefore also by a finite number of closed intervals [a',~ ; b',] (2 (ax ; b~)

d~ with no common endpoints. In each of these [a'~ ;b'~] ~ is in

LP(a',~;b'~); heace it is in L P ( a ; b ) .

Remark . The sufficient condition for absolute continuity of ~ in ( a ;b )

given in the last theorem, i.e. that o ( x ) < 1 in ( a ; b), is certainly not

necessary. It is easy to construct a function q ) ( ~ ) = f O (~) d~ ~. _ ( w i t h a

measure d B ( ~ . ) - - - - 0 Q 0 d ) . ~ t ( 2 , 2 ) , hence #() . )~L~(--~;~) , such that

for the corresponding f , m ( f ; x ) = 1 for all x. It is enough to take any

0(~.) in L ~ ( - - ~ ; ~ ) which is not in LP(a;b) for any p > 1 and any

interval (a ; b).

We shall close the section by a simple characterization of the case

of purely singular measure g.

T h e o r e m IX. I n o r d e r t h a t t h e m e a s u r e ~ b e s i n g u l a r

i t is n e c e s s a r y a n d s u f f i c i e n t t h a t f (x ) b e e q u i v a l e n t t o a

c h a r a c t e r i s t i c f u n c t i o n o f a m e a s u r a b l e s e t ,


P r o o f . ~t is s ingular i f and on ly i f any m i n i m a l s u p p o r t o f I~t ~ is o f

Lebesgue m eas u r e O. By u s i n g the m i n i m a l s u p p o r t ~0 i n t r o d u c e d at the

d F (x) b e g i n n i n g o f th i s sec t ion we get the c o n d i t i o n f ( x ) - - dx = 0 or 1

a l m o s t everywhere , w h e n c e the t h e o r e m .

B I BL 1 0 G R A P H Y

I t [ A c h i e z e r, N. I. and K r e i n M. G., On certain problems in the theory of moments, Kharkoff (1038) (Russian).

[2] A r o n s z a j n, N., On a problem of Weyl in the theory of singular Sturm- Liouville equations. Am. Journ. of Math. 79 (1957), pp. 597--610.

[2'] A r o n s z a j n. N., Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues 1. Operators in a Hilbert space. Proc. of Nat. Ac. $ci. 34 (1948) pp. 474--480.

[3] B e s i c o v i t c h , A.S., On a general metric property of summable functions, JotLrn. London Math. Soc. vol. I (11126), pp. 120--128.

[4] D e I a n g e, H., On two theorems of S. Verblunsky. Proc. Camb. Phil. Soc. vol. 46 (1050) pp. 57--66.

[5] D o o b, J. L., and K o o p m a n, B.O. , On analytic functions with positive imaginary parts, Bull. Amer. Math. Soc. vol. 40 (1934) pp. 601--805.

[6] E v a n s, G . C . , The Logarithmic Potential. Am. Math. $oc. Colloquium Publications vol. VI 0927).

[7] G e l f a n d , I. M. and L e v i t a n , B. M., On the determination of a differential equation from its spectral function, hvestia Akad. Nauk USSR Set. Math. vol. 15 (1951) (Russian).

[8] K a t z, I. S., On the integral representation of analytic functions mapping the upper half-plane into itself, Uspehi Math. Nauk vol. I l 3 (69), (1950), pp. 130--144 (Russian).

[9J K o l m o g o r o f f , A., Sur les fonctions harmoniques conjugu~es et les shries de Fourier. Fund. Math. vol. 7 (1925) pp. 24--29.

[10] K r e i n, M. G., On a method of effective solution of an inverse boundary problem. Dokl. Akad. Nauk USSR 94 (1054) p. 420 (Russian).

[11] K r y l o f f , W., Ueber Funktionen die in de: Halbebene regul~ir sind. Math. Sbornik N. S. vol. 6 (19.~8) pp. 95--138 (Russian).

[12] N e v a n l i n n ~ , , F. and N i e m i n e n , T., Das Poisson-Stieltjes'sche Integral und seine Anwendung in der Spektraltheorie des Hilbert 'schen Raumes. Ann. Acad. Sci. Fennicae 207 (l:~r)5) pp. 1--38.

[13] N e v a n l i n n a, R o I f, Eindeutige Analytische Funktionen, Berlin 1936. [14] P l e s s n e r, A., Zur Theorie der conjugierten trigonomentrischen Reihen.

Giessen, 1022. [15] P o I I a r d, S., Extension to Stieltjes Integrals of a theorem due to Plessner.

Journ. London Math. Soc. vo l 2 (1027) pp. 37--41. [16] R i e s z , M,, Sur les fonctions conjuguhes. Math. Zeit. Vol. 27 (1928),

pp. 218--244. [17] T i t c h m a r s h , E. C., On Conjugate Functions. Proc. London Math. Soc.

Vol. 29 (1920) pp. 4~,~--8o.


| t 8 ] V e r b l u n s k y , S., Two moment problems for bounded functions. Proc. Camb. Phil. Soc. Vol. 42 (1946) pp. 189--19~;.

[19] V e r b l u n sky , S., On the initial moments of a bounded function. Proc. Camb. Phil. Soc. Vol. 43 (i~47) pp. 275--279.

[20] W o l f f , J u l i u s and de K o c k , F., Les fonctions holomorphes ~i pattie r6eUe positive et Yint6grale de Stieltjes. Btdl. Soc. Math. France Vol. 60 (19.';2)pp. 221--227.

Department of Mathematics University of Kansas Lawrence, Kansas, U.S.A.

(Received October 10, 1957)

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On exponential representations of analytic functions in the upper half-plane with positive imaginary part