PERMUTATIONS OF TAYLOR COEFFICIENTS OFBOUNDED FUNCTIONS

BY WALTER IUDIN

1. Introduction. In this paper, a will always denote a function whose domainof definition is a set Y of non-negative integers and whose values are non-negativeintegers. To avoid trivialities, Y will always be assumed to be an infinite set.Every such a induces a linear transformation I, in the set of all formal powerseries" if

I(z) c( )z

I,] is defined by

c(.On))zmeY

If Q and R are two classes of power series, and if 9] R for all ] Q, we saythat a carries Q to R. If a carries Q to Q, we say that a preserves Q.The main result of this paper is a complete characterization of those functions

a which preserve the class of all bounded analytic functions in the open unitdisc. Before stating this result, it may be of some interest to point out that itis often quite easy to determine the functions a which preserve a class whichis characterized by conditions which bear only on the absolute values of thecoefficients. Let us illustrate this with a few examples; the proofs are so ele-mentary that we omit them:

(a) Let Pr be the class of all power series whose radius of convergence is atleast r. Let D* and D. be the upper and lower limits, respectively, of a(m)/m,asm--. inY. Then

(i) a preserves P if and only if D. > 0;(ii) for 1 < r < , a preserves Pr if and only if D. >__ 1;(iii) a preserves P1 if and only if D* <(iv) for 0 < r < 1, a preserves Pr if and only if D* <_ 1.

(b) Let H be the class of all power series with ’ c(ra) [ < a preservesH if and only if there is an integer k such that no integer has more than k inverseimages under a. The same result holds for all functions with finite Dirichletintegral in z < 1, and for the class of all power series which converge absolutelyonlz[ 1.

It is convenient to make the following definition: If S is an infinite arithmetic

Received April 4, 1961. Sponsored by the United States Army under Contract No.11-022-0RD-2059. Mathematics Research Center, United States Army, Madison, Wisconsin,and by NSF Grant G-14362.

537

538 WALTER RUDIN

progression in Z/ (where Z/ is the set of all non-negative integers), say S{/o,/o-k p, lc-k2p, ...},ifaZ/,bZ+,and

(3) a(/c -{- jp) a -k jb (j 0, 1, 2,...),

then we call a an ane map o] S into Z+.For our later work it is important to observe that a is affine if and only if

its graph is an arithmetic progression in the plane.

THEOREM. Let B be the class o] all bounded analytic ]unctions in the open unitdisc U, and let A be the class o] all ] B which are uni]ormly continuous in U.Each o] the ]ollowing conditions on Y and a implies the other three:

(I) a preserves A.(II) a carries A to B.

(III) a preserves B.(IV) There are disjoint infinite arithmetic progressions $1 Sr in Z/

and there are non-constant ane maps a o] S into Z+, such thatY differs ]rom $1 k.) L) S in at most finitely many places, (b) onY S a coincides with a

The theorem will be proved via the following cycle of implications" IVIII --+ I -- II --. IV.

2. Remarks. As is known, property IV (with the set Z of all integers in placeof Z/ and with $1 S, infinite in both directions) characterizes the mapsa which preserve L (see [5]; the result was extended to arbitrary locally compactAbelian groups by Cohen [2]; a complete discussion may be found in [7; ChaptersIII, IV]) and also characterizes the maps a which preserve C and L, the spacesof all continuous and bounded measurable functions on the circle, respectively([7; Theorem 4.6.9]). The fact that we now deal with the semigroup Z/ ratherthan with all of Z introduces a new element into the problem.To illustrate this, consider the space H1, the set of all analytic functions f

in U for which

is a bounded function of r(0 _< r < 1); H may be identified with the space ofall f t L whose Fourier coefficients vanish on the complement of Z/. If a

satisfies IV, then a preserves H. But the converse is false. For if , > 1 and{nk} is a sequence of positive integers such that nk/l/nk > and if /(z)

c(m)z is in H, then [4] c(nk) < . Hence if a(m) m for all m g {nkand if the restriction of a to {nk} is an arbitrary one-to-one transformation of{nk onto Ink }, then a preserves H. In fact, a is a one-to-one map of H ontoH, and uncountably many are obtained in this way, whereas only countablymany a satisfy IV.

PERMUTATIONS OF TAYLOR COEFFICIENTS 539

3. Proof that IV --+ III --+ I --+ II. We suppose first that Y consists of a singlearithmetric progression, Y {k, k + p, k -t- 2p, and that

(3) a(k + jp) a + jb

If ](z) _ffo c(m)z", ] . B, and

(j= 0, a, 2, ...).

(4) g(z) _, c(a -at- jb)zk+iE,

we have to prove that g B. Put

1 Efl(zr), f3(z) f(zl/b),(5) fl(z)m-O-’ c(a "k" m)z, f.(z)

r-

where z zb are the b distinct b4h roots of z (z 0).differs from ](z) by a polynomial, f B. Since

Since Zafl (Z) differs

(6) ].(z) _, c(a + jb)z’,=0

it follows that ] and ]. belong to B, and since

(7) a(z) zl(z),the same is true of g.

Finitely many applications of this result show that IV implies III.Note that A and B are Banach spaces, with the norm

(s) Ill sup ](z) I.If III holds, the closed graph theorem shows that I, is a continuous linear

transformation of B into B. Suppose that for some n Z+ there are infinitelymany m such that a(m) n; taking f(z) z, we then have ()(z) zwhich is not in B, and we have a contradiction. Thus a is finite-to-one. Infact, if n Z+, then a-i(n) contains no more than [I I, II elements.

In particular, I,] is a polynomial whenever ] is a polynomial. Since A is theclosure, in the norm (8), of the set of all polynomials, the continuity of impliesthat I,] A if A. Thus III implies I.

It is trivial that I implies II.Our proof that II implies IV requires a rather detailed knowledge of the

idempotent measures on the torus; it will be completed in 5.

4. Indempotent measures on the torus. If t is a (complex, bounded, Borel)measure on the two-dimensional torus T2, its Fourier-Stieltjes transform isdefined on Z2, the direct sum of Z and Z (i.e., the set of all lattice-points in theplane), by

(9) p(m, n) e-(’+") dry(x, y) (m Z, n . Z).

540 WALTER RUDIN

If $ p we call t idempotent; it is clear that then 0 and 1 are the only possiblevalues of p, and we define S() to be the set of all (m, n) Z at which p(m, n) 1.

It will be convenient to refer to a coset of a subgroup of Z simply as a coset inZ. A coset of a cyclic subgroup will be called a cyclic coset (it is nothing buta doubly-infinite arithmetic progression). The coset-ring of Z is the smallestfamily t of subsets of Z which has the following properties: every coset in Zbelongs to gt, and 12 is closed under the formation of finite unions, finite inter-sections, and complements.The idempotent measures are known explicitly ([6]; see [1] for the generaliza-

tion to arbitrary compact Abelian groups):

4.1. A set S C Z is S() /or some idempotent measure t on T i] and onlyi] S() belongs to the coset-ring o] T.Thus each S() is the disjoint union of finitely many sets of the form

(10) M ( L:.

where M is a coset in Z2, and the L are complements of cosets L Further-more, we may arrange it so that each L is a subgroup of M’, and that the indexof L in M’ is infinite, unless M’ {0 }. (Here M’ and L are the groups whichhave M and L as cosets.)

It is now easy to deduce the following:

4.2. is a continuous lidempotent measure on T i] and only i] S(t) is thedisjoint union o]finitely many sets o] the form (10), with M a cyclic coset or a singlepoint.

Our next lemma concerns singular measures, i.e., measures which are con-centrated on sets of tIaar measure 0.

4.3. I] is a continuous singular idempotent measure on T2, then S() is thedisjoint union o] finitely many cyclic cosets in Z.

In particular (and this is the application we need) if A is a cyclic coset suchthat A ( S() is infinite, then A C S().

Proo]. By 4.2, S() differs from A A a finite union of cycliccosets, by at most a finite set. If S() A and if a , then a is singu-lar and differs from p at no more than finitely many points of Z. Hence

is absolutely continuous. Since is singular, it follows that . Iftwo of the sets A had a point in common, then p would be at least 2 there,since p , which is false.

4.4. Suppose is a measure on T, and {q} is a sequence in Z which tendsto infinity, such that the limit

(11) (p) lim p(p -{- q)

PERMUTATIONS OF TAYLOR COEFFICIENTS 541

exists ]or every p Z2. Then is the Fourier-Stieltjes transform o] a singularmeasure on T2.

This is the two-dimensional version of a lemma of Helson [3]; it remains trueif T is replaced by any compact Abelian group [71 Lemma 3.5.1].

4.5. Suppose {c.} is a sequence o] Fourier-Stieltjes coe2cients (of a measureon the unit circle), and c, 0 or 1 ]or all n

_O. Then there exist positive integers

no and r such that c/r c for all n

_no

This is also due to Helson [3].

5. Proof that II implies IV. We associate with every trigonometric poly-nomial F on T of the form

(12) F(x, y)0

(only finitely manyc are different from O) an ordinary polynomial F, by seting

(13) F(e’ F(x, y).

If II holds, the closed graph theorem shows that is a bounded linear operatoron A, and the map

f

is therefore, for each x, a bounded linear functional on the space of all trig-onometric polynomials (12), of norm ]] [. So is the map

By ghe Hahn-Banaeh gheorem, (1) can be exgended go a bounded linear fune-gional on (T), and ghe ies represengagion gheorem ghen assergs ghe exisgenee

of a measure

a (z,

for all trigonometric polynomials F of the form (12). Taking

(17)

we have

(18)

and (16) shows that

(19)

F(x, y) e

(F)(e’) e-, ek,a(k)n

d(m,n) =1 if (m)-n,

[o otherwise.

(m Z, n Z/),

542 WALTER RUDIN

Let E be the graph of a, i.e., the set of all points in Z of the form (m, a(m)),and call the set of all (m, n) Z with n > 0 the upper hall-plane. Equation (19)can then be stated as follows:

I/’ II holds, there is a measure on T such that the restriction o] d to the upperhall-plane is the characteristic ]unction o] E.

Let us now assume (this will lead to a contradiction) that E cannot be coveredby any finite collection of cyclic cosets in Z. If A A2, A, is an orderingof all cyclic cosets, it follows that for i 1, 2, 3, there exists p (m, n) Eso that p, A1 k.) k.) At Since o-i(n) contains at most !1 II points (seethe proof that III implies I), no horizontal line intersects E in more than [i I, !1points. Hence n --. o. By taking a suitable subsequence of {p}, which weagain denote by {p}, we can force the existence of

(20) (p) lim d(p + p,)

for every p Z.By 4.4, o p, where/z is a singular measure on T. Since n -- o and since

d 0 or 1 in the upper half-plane, is idempotent. No horizontal line intersectsS(g) in more than II I, [[ points. Hence 4.1 (and the remarks following it) showsthat S(g) is a finite union of sets (10) in which M is cyclic or M’ {0}, andthus is continuous. Aslo, (0) 1, since d(p) 1 for i 1, 2, 3, andso 0.We can now conclude from 4.3 that S(t) contains a cyclic group A, generated

by g (m0, no) with no > 0.Returning to our sequence {Pi}, it follows that E contains the points

(21) p pi "- g, ", p "- ag,

where {a} is a sequence of positive integers which tends to . Since E hasonly finitely many points on each horizontal line, we see that for every sufficientlylarge i ther is an integer b

__0 such that the points p bg, with 0 <_ b

_b

are in E, while p (b -- 1)g is in the upper half-plane but is not in E.Put q p bg, and repeat the preceding construction, with {q} in place

of {p }. The construction then yields a continuous singular idempotent measureon T such that the points ag are in S(},) for a 0, 1, 2, but -g S(h).

This contradicts 4.3.We have now proved that E is covered by a finite collection of cyclic cosets

in Z. If A; is one of these cosets, we may assume that A {p - iq}, wherei Z, and p and q are in the upper half-plane. Define

(22) c, d(p z nq) (n . Z),

where z is the measure for which (19) holds. Then {c.} is a sequence of Fourier-Stieltjes coefficients, and for n > 0, c, is either 0 or 1. By 4.5, {c,} is ultimatelyperiodic, as n -- . This means that A; t’N E differs from the union of finitelymany arithmetic progressions in at most finitely many places.

PERMUTATIONS OF TAYLOR COEFFICIENTS 543

Since E (A1 E) k (Ak E), a finite union, and since E is thegraph of a, it follows that a satisfies IV. This completes the proof.

REFERENCES

1. P. J. COHEN, On a conjecture of Littlewood and idempotent measures, American Journal ofMathematics, vol. 82(1960), pp. 191-212.

2. P. J. COHEN, On homomorphisms of group algebras, American Journal of Mathematics, vol.82(1960), pp. 213-226.

3. HENRY HELSON, On a theorem of Szeg, Proceedings of the American Mathematical Society,vol. 6(1955), pp. 235-242.

4. R. E. A. C. PALEr, On the lacunary coecients of power series, Annals of Mathematics, vol.34(1933), pp. 615-616.

5. WTER RUDIN, The automorphisms and endomorphisms of the group algebra of the unitcircle, Acta Mathematica, vol. 95(1956), pp. 39-56.

6. WALTER RUDIN, Indempotent measures on Abelian groups, Pacific Journal of Mathematics,vol. 9(1959), pp. 195-209.

7. WALTER RUDIN, Fourier Analysis on Groups, Interscience Publishers (in press).

UNIVERSITY OF WISCONSIN