A Fourier series method for meromorphic and entire functions

B ULLETIN DE LA S. M. F.

L.A. R UBEL

B.A. T AYLOR

A Fourier series method for meromorphic and entire functions

Bulletin de la S. M. F., tome 96 (1968), p. 53-96

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Bull. Soc. math. France, 96, 1968, p. 53 a 96.

A FOURIER SERIES METHOD

FOR MEROMORPHIC AND ENTIRE FUNCTIONS

LEE. A. RUBEL (*) AND B. A. TAYLOR

University of Illinois.

Contents.

Pag-es.

I n t r o d u c t i o n . . . 54

1. An analysis of sequences of complex n u m b e r s . . . 56

1.1. Definition : n (r, Z ) . . . 57

1.2. Definition : N (r, Z). . . . 57

1.3. Proposition : N(r, Z) = V log(r/ [ zj). . . . 57

1.4. Proposition : n (r, Z) = r (dfdr) N (r, Z ) . . . 57

1.5. Definition : 5 ( r ; A - : Z ) . . . 57

1.6. Definition : S (r,, 7*3; A-: Z). . . . 57

1.7. Definition : Growth f u n c t i o n . . . 67

1.8. Definition : Finite A - d e n s i t y . . . 58

1.9. Proposition : n (r, Z ) ^ A A (5r). . . . 58

1.10. Definition : ^ - b a l a n c e d . . . 58

1.11. Proposition : Finite density and A-balanced implies strongly ^ - b a l a n c e d . . . 58

1.12. Definition : ^ - p o i s e d . . . . . . 5g 1.13. Proposition : Finite ^-density and A-poised implies strongly ^-poised. 59 1.14. Proposition : A-balanced if and only if ^-poised. . . . 5g 1.15. Definition : ^ - a d m i s s i b l e . . . 60

1.16. Propsition : Conditions for A-admissibility . . . 60

1.17. Definition : Remainder. . . . 61

1.18. Definition : Complete set of r e m a i n d e r s . . . 61

1.19. Theorem : Existence of complete set of remainders. . . . 61

1.20. Lemma : Convex functions, j —7/ . ^ ) . . . 61

2. The Fourier coefficients associated -with a s e q u e n c e . . . 65

2.1. Definition : S^r; k : Z ) . . . 65

2.2. Proposition : | 5"(r; k : Z) [ ^ (A--1) N (er, Z ) . . . 65

2.3. Definition : c^(r; Z : a). . . . 65

2.4. Definition : ^-admissible. . . . 65 (*) The research of the first author was partially supported by the United States Air Force Office of Scientific Research, under Grant Number AFOSR 460-63..

o4 L. A. RUBEL AND B. A. TAYLOR.

Pages.

2.5. Proposition : A-admissible if and only if there exist ^-admissible

sequences of Fourier coefficients. . . . 65

2.6. Proposition : Condition for X-admissibility of { c^r; Z : a) } . . . 66

2.7. Definition : E^r; Z : a ) . . . 67

2.8. Proposition : { c ^ ( r ; Z : a) } are ^-admissible if and only if E,(r; Z a)^ A^(Br)... 67

2.9. Theorem Existence of ^-balanced complete set of remainders... 67

2.10. Corollary lim Eg^; Z' (J?): a(2?)) == o . . . 69

3. Sequences that are A - b a l a n c e a b l e . . . 69

3.1. Definition ^ - b a l a n c e a b l e . . . 69

3.2. Definition Regular. . . . 69

3.3. Proposition : A-admissibility in case ^(r) = r ? . . . 69

3.4. Definition Slowly increasing. . . . 71

3.5. Proposition : Slowly increasing implies regular. . . . 71

3.6. Proposition : log ^(e-") convex implies A is r e g u l a r . . . 71

3.7. Lemma : Condition that )v be slowly i n c r e a s i n g . . . 71

4. The Fourier coefficients associated "with a meromorphic function.. . . . 76

4.1. The Nevanlinna characteristic. . . . 76

4.2. Lemma : Expressions for the Fourier c o e f f i c i e n t s . . . 76

4.3. Definition : Finite > v - t y p e . . . 77

4.4. Definition : A g . . . 77

4.5. Proposition : Condition that an entire function be of finite A-type... 77

4.6. Theorem : Conditions on the Fourier coefficients that a mero- morphic function be of finite ^ - t y p e . . . 78

4.7. Theorem : Conditions on the Fourier coefficients that an entire function be of finite A - t y p e . . . 80

4.8. Definition : E^ (r, f ) . . . 82

4.9. Theorem : E^(r, f) ^ A).(Br). . . . 83

4.10. Theorem : / exp ^ i + £ . . . 83

5. Applications to entire f u n c t i o n s . . . 84

5.1. Theorem : Existence of entire functions with prescribed Fourier c o e f f i c i e n t s . . . 84

5.2. Theorem : Conditions that a sequence be the sequence of zeros of an entire f u n c t i o n . . . 87

5.3. Theorem : Conditions that a sequence be the sequence of zeros of a meromorphic function. . . . 88

5.4. Theorem : A == A^/A^ if and only if A is regular. . . . 90

5.5. Lemma : log M(r, f) ^ 3 E^{i r, f). . . . 91

5.6. Lemma : lim g^ (z) = i . . . 91

5.7. Theorem : Generalized Hadamard product. . . . 92

Appendix : Delange's proof of theorem 5 . 2 . . . 93

R e f e r e n c e s . . . 96

Introduction. — This work is a further development and application of the Fourier series method for entire functions introduced by the first author in [5]. The idea which was presented there and is exploited

further here is the following : if fis a meromorphic function in the complex plane, and if

c,(r, f) = ^ f\log | /-(re-9) |) 6-^d9

«- —7t

is the /c-th Fourier coefficient of log | f(re^) |, then the behaviour of f(z) is reflected in the behaviour of the sequence j c^(r, f)} and vice versa.

We prove a basic result in theorem 4.6, which characterizes the rate of growth of f in terms of the rate of growth of the c^(r, f) and the density of the poles of f, generalizing theorem 1 of [5]. We apply this theorem, as in [5], to obtain estimates for some integrals involving | f (z) \ and to obtain information about the distribution of the zeros of an entire function from information about its rate of growth.

By these means, we make a study of certain general classes of mero- morphic and entire functions that include many of the classically studied classes as special cases. Let ^(r) be a positive, continuous, increasing, and unbounded function defined for all positive r. We say that the mero- morphic function f is of finite ^-type to mean that there exist positive constants A and B with T(r, f)^A^(Br) for r > o, where T is the Nevanlinna characteristic. An entire function f will be of finite ^-type if and only if there exist positive constants A and B such that

f(z) | ^ exp (A ^ (B \ z ])) for all complex 2.

If we choose ^ (r) = rP, then the functions of finite 7-type are precisely the functions of growth not exceeding order p, finite exponential type.

We obtain here complete answers to certain basic questions about func- tions of finite ~>-type. For example, we characterize, in theorem 5.2, the zero sets of entire functions of finite 7-type. This generalizes the well-known theorem of Lindelof that corresponds to the classical case ~h (r) == rP. We obtain, in theorem 5.3, a corresponding result for meromorphic functions. Then, in theorem 5.4, we give necessary and sufficient conditions on ^ that each meromorphic function of finite

^-type be the quotient of two entire function of finite ^-type.

Further, we obtain, in theorem 5.7, a ^(< generalized Hadamard product"

for entire functions of finite ^-type. It serves many of the same purposes as the Hadamard canonical product, and is considerably more general.

In particular, if X satisfies some additional conditions, and if fis an entire function of finite ?i-type, then there will be an unbounded set ^ of positive numbers R, and corresponding entire functions fp of finite ?i-type, such that the zeros of /p are the zeros of f in the disc { z : \ z \ ^ R} and such that /R—^ fnot only uniformly on compact sets, but also in a way consistent with 7. We call the sequence ( /p } the generalized Hadamard product.

This result has been used in an essential way by the second author [6]

in proving that spectral synthesis holds for mean-periodic functions in certain general spaces of entire functions.

The body of the paper is divided into five parts, the last two of which contain the main results. The first three sections are concerned with various elementary, although sometimes complicated, results on sequences of complex numbers. The first section discusses the distribution of these sequences. The <( Fourier coefficients " associated with a sequence are defined in the second section and several technical propositions involving these coefficients are proved there. The third section is concerned with the property of regularity of the function ^, which is closely connected with the algebraic structure of the field of meromorphic functions of finite ^-type. The fourth section contains the generalizations of the results of [5]. Finally, in the fifth section, the results about the distri- bution of zeros, and the generalized Hadamard product are proved.

We urge that on a first reading, the reader read § 4 first and then § 5, referring to § 1, § 2, § 3 for the appropriate definitions and statements of necessary preliminary results. After this, the complex sequence theory of the first three chapters will seem much more natural.

We shall use, for a function 9(r), the notation 0(^(r)) to denote a func- tion that is bounded in modulus by Acp(r) for some constant A, and the notation 0(^(0(r))) to denote a function that is bounded in modulus by A cp (Br) for some constants A and B.

It seems clear that through much of this paper, the assertions about entire functions can be replaced by corresponding assertions about subharmonic functions, and the assertions about meromorphic functions can be replaced by corresponding assertions about the differences of subharmonic functions, without requiring any real change in the proof.

It is by now a standard procedure to replace the logarithm of the modulus of an entire function by a general subharmonic function, replacing the zeros of the entire function by the masses that occur in the Riesz decompo- sition of the subharmonic function. For numerous reasons, though, we have preferred to keep this paper in the context of entire and mero- morphic functions.

1. An analysis of sequences of complex numbers.

We study here the distribution of sequences Z== { Zn}, n == i, 2, 3 , . . . , with multiplicity taken into account, of non-zero complex numbers z//, such that Zn -> oo as n --> oo. Such sequences Z are studied in relation to so-called growth functions ^. We denote by A and B generic positive constants. The actual constants so represented may vary from one occurrence to the next. In many of the results, there is an implicit uniformity in the dependence of the constants in the conclusion on the

constants in the hypotheses. For a more detailed explanation of this uniformity, we refer the reader to the remak following proposition 1.11.

Let Z == { Zn { be a sequence of non-zero complex numbers such that lim Zn == oo as n -> oo.

1.1. DEFINITION. — The counting function of Z is the function n(r,Z)= ^ i.

I ^-n I ^ r

1.2. DEFINITION. — We define

^nd, Z),, N(r,Z)== -^di.

J Q 1.3. PROPOSITION. — We have

N(r,Z)= ^ ^g^T

^™ I ^ | Proo/*. — Note that

2 l o g — — = r i o g ^ ) d [ n ( ^ Z ) ] .

\^r [ z n { Jo v /

The proposition follows from an integration by parts.

1.4. PROPOSITION. — We have

n ( r , Z ) = r ^ N ( r , Z ) . Proof. — Trivial.

1.5. — DEFINITION. — We define, for k = i, 2, 3, . . . and r ^ o,

^.^-4 2 (, ¹ )'.

\^-n\-=r

1.6. DEFINITION. — We define, for k == i, 2, 3, . . . and fi, fs^o, S(ri, r.2; k: Z) = 5(r.2; k : Z) — 5(fi; A:: Z).

When no confusion will result, we will drop the Z from the above notation, and write n (r), S (r; k), etc.

1.7. DEFINITION. — A growth function ^(r) is a function, defined for o < r < oo, that is positive, non-decreasing, continuous, and unbounded.

Throughout this paper, A will always denote a growth function.

58 L. A. RUBEL AND B. A. TAYLOR.

1.8. DEFINITION. — We say that the sequence Z has finite ^-density to mean that there exist constants A, B such that for all r > o,

N(r, Z)^A7(Br).

1.9. PROPOSITION. — If Z has finite ^-density, then there are constants A, B such that

n(r, Z)^A^(Br).

Proof. — We have

n(r, Z) log2 ^ r7 "^ d/^B (27-, Z).

1.10. DEFINITION. — We say that the sequence Z is ^-balanced to mean that there exist constants A, B such that

( l . l O . z ) l ^ r . r ^ ^ ^ l ^ - A ^ + A ^

for all r-i, r.2 > o and k := i, 2, 3, . . . . We say that Z is strongly ^-balanced to mean that

(1 10 2^ S<r r ' k ' 7\ I ^ A ^(^i) , A ^(BrQ H.iu.2^ A(ri., 7-2, / c . Z ) [ ^ —^— + —^—

for all Fj, r.2 > o and k = i, 2, 3, . . . .

1.11. PROPOSITION. — If Z has finite ^-density and is ^-balanced, then Z is strongly ^-balanced

Remark. — Using this result for illustrative purposes, we make explicit here the uniformity that we leave implicit in the statements of similar results. The assertion is that if Z has finite ^-density, with implied constants A, B, and is ^-balanced with implied constants A', B ' , then Z is strongly ^-balanced with implied constants A", B" that depend only on A, B, A ' , B' and not on Z or ^.

Proof of 1.11. — We observe first that, if r > o, and if we let r ' ' = r^^

then

( l . l l . i ) S(r,r'; /Ol^³^?.

To prove this, we note that

S ( r , r ' ; k - ) ^-f'-dn(t),

v r

from which ( l . l l . i ) follows after an integration by parts. Now, for 7-1, r-2 > o, let r\ = r,^^ and ^ == 7*2 k1^. Then

| S(r,, r,; k) | ^ 5(r,, r,; A:) [ + | S(ri, r',; A-) + I 5(r2, ^; /c)

On combining this inequality with (l.ll.i), (1.9), and the fact that ]^/k^ ^ ^Q have

S(r,, r,; k) [ ^ 1 S(r\, r.; /c) + ^A^r,) + ^A7(Br,).

But, by hypothesis,

| S(r\, r',; k) ^ ^ A ^(Br,) + ^ A ^r,) for k === i, 2, 3, . . . .

1.12. DEFINITION. — We say that the sequence Z is t-poised to mean that there exists a sequence a of complex numbers a == } a/: }, A: == i, 2, 3,...

such that, for some constants A, B, we have, for /c == i, 2, 3, . . . and r > o, A^CBr)

T ^k ' (1.12.1) | a , + ^ ( r ; ^ : Z )

If the following stronger inequality

(1.12.2) ^ + S ( r ' , k : Z ) ^^^_^k holds, we say that Z is strongly '}-poised.

1.13. PROPOSITION. — If Z has finite ^-density and is ^-poised, then Z is strongly ^-poised.

Proof. — The proof is quite analogous to the proof of 1.11, based on the substitution r ' = r/c1^. We omit the details.

1.14. PROPOSITION. — A sequence Z is ^-balanced if and only if it is ^-poised, and is strongly ^-balanced if and only if it is strongly ^-poised.

Proof. — We prove only the second assertion, since the proof of the first assertion is virtually the same. If it is first supposed that Z is strongly 7-poised, where { a / : } is the relevant sequence, then we have

| S(r,, r,-, k) | = S(r,; k) + ^—^—S(r,', k)

^\^+S(r;k)\+\^+S(r,;k) so that Z is strongly ^-balanced.

Suppose now that Z is strongly 7-balanced, with A, B being the relevant constants. Let

p(7) = inf \ p = i, 2, 3, . . . : lim inf ^ = o I

( r>°° rf )

Naturally, we let p (^.) == oo in case lim inf )\ (^)|r^p > o as r^ > oo for each positive integer p. For i^k<.p(^), we have inf r-^ 7 (Br) > o

6o L. A. RUBEL AND B. A. TAYLOR.

for r > o. Thus, there exist positive numbers r/c such that HBrk) A(Br)

n

-^-

for r > o and i ^k < p(^). For k in this range, we define (1.14.1) ^==—S(r,;k).

For those k, if there are any, for which k ^p(^), we choose a sequence o < pi < 0.2 < . . . with p; ->ooasj-^oo, such that

T ^CBP/)

/S-p-

-

For values of A-, then, such that k ^ p (/), we define (1.14.2) ^ = l i m — 5 ( p / ; A : ) .

/ - ^ a o

To show that the limit exists, we prove that the sequence i S ( o / ; k) ;, j = i, 2, . . . , is a Cauchy sequence. Let

We have

^ = ^(P/; ^) — 5(p,,; A-) == 5(p,,, p;; A:).

| A . i^-^^^P^) , A/(Bp;)

I ^•y,/^ I := ——,—,—— + ——-—,——- K^m A:?)

Since p^ p/^) for p ^ i, it follows from the choice of the py that Ay.,,, -^ o as j, m -> oo. We now claim that

I^+^A)!^3^^.

— A:^

For, if i ^k< p(/i), then

a.+ ^(r; A) | = | ^fa r; k) \ ^ A^B^ + ^W ^ ^W kr^ ^ — kr^

If k ^ p (/), then

[a,+S(r;/c)|=lim 5(r, p,; ^ | ^AA(Br) + lim sup A A ( B^ = A^Br).

/-^- ^/l /-^oo " /cp) A•r^• 1.15. DEFINITION. — We say that the sequence Z is ^-admissible to mean that Z has /zm7e ^-density and is ^-balanced.

In view of propositions 1.11 and 1.13, the following result is immediate.

1.16. PROPOSITION. — Suppose that Z has finite ^-density. Then the following are equivalent :

(i) Z is ^-balanced;

(ii) Z is strongly 7-balanced;

(iii) Z is t-poised;

(iv) Z is strongly 1-poised;

(v) Z is ^-admissible.

In proposition 3.3, we give a simple characterization of /-admissible sequences in the special case A (r) == r?.

We next consider the effect that deleting from Z those finitely many terms that lie in the disk { z : \ z | ^ R} has on S(r,, r.,; k).

1.17. DEFINITION. — We define Z (R), for J? > o, to be the sequence obtained by deleting from Z those terms of modulus not exceeding R.

That is,

Z{R)=Zr\[z:\z > j R ; , and we call Z (R) the R-remainder of Z.

1.18. DEFINITION. — Let (^ be a non-empty set of positive real numbers.

The collection of remainders { Z (J?) : E e ^ j is caZ/ed complete if ^ is unbounded.

1.19. THEOREM. — Each strongly ^-balanced sequence Z has a strongly 7-balanced supersequence Z' such that n (r, Z ' ) = 0 (n (r, Z)) and such that Z' has a complete set of remainders that are uniformly strongly A-balanced. In the special case in which lim inf r-° 7 (r) == 30 for each

f-^x

positive number p, we may take Z' == Z. Jn ^e special case in which l o g A ^ ) is convex, we may take Z'=Z, and we may take the collection of remainders to be { Z(R) == R ^ R,} for some number J?o.

Before proving theorem 1.19, we derive first some elementary facts about the behaviour of the functions r~^^(r) when A has a particularly nice form. In the following, we will denote by u (x) the function defined for — oo < x < co by u(x) = log /(e^). We observe that u is a non- decreasing function.

1.20. LEMMA. — Suppose that u(x) == log A e^) is convex and that (T is a positive number. Then the function /—^(r) decreases to its infimum as r increases, and increases thereafter. If for some positive o- we have lim sup /'-^(r) < GO, then there exists a constant M such that

7->^

/(sr) f^M^(r). Further, there is a positive number Ro such that for every R ^Ro, there exists a positive number 0-== o-(J?) such that

W -inf^)

R^^ 7^

Proof. — Since u is convex and increasing, we may write (x)==u(o)+ rh(f)dt,

^ n U (X) == U (0

where h is a non-negative and non-decreasing function. If x == log r, then

^) = exp ju(o) + F(h(t)-a) dt\,

{ ^ 0 ;

from which the first assertion follows immediately. To prove the second assertion, observe that if lim sup —7/r^v^/ << oo, then h(t) must be bounded- say h (0 ^ C. Then

-^—⁾ = exp ; u(x + log 2)— u(:r) j ^ e x p { C log 2 ;.

To prove the last assertion, let Xo==\ogR and a == h(xo). Since A is non-negative, non-decreasing, and not identically zero, it follows that

<7 is positive if R is sufficiently large. Then

...X-

(u(x)—ax)—(u(x,)—crx,)= ^ (A(0—cr)d^o.

x

Hence,

^CR) / / . . . , , , . . p Ur)

—— = exp j u(Xo)—axo} =mf exp j u(x)—ax==im —^-'

JL\ x /•>0 f

Proof of theorem 1.19. — By hypothesis, there exist constants A and B such that

(1.19.,) |Sfr,,,,;t:Z)^^+A^.

' 1 ^ 2

Let J? and o- be positive numbers for which HBR)_ HBr)

R- >o ^ • We claim that then

H 1Q ^ I Sfr r - k • Zf7?^ I ^ ^^^^ + 2A^Br2)

^i. iy. 2; \ ^ (Ti, r.2.., / c . L, {n)) \ ^ —-j^— -t- —^—

If ri ^r.2 ^R, then S(r^ r.2; k : Z(R)) == o and there is nothing to prove.

If R ^ Fi ^ r,, then

5(ri, r,; / c : Z(jR)) = S(ri, r,; A : Z)

and so (1.19.2) follows from the inequality (1.19.i). If r^R^r.^

then

| S(r,, r,; k : Z(R))\ = | S(R, r,; k : Z) [, which, by (1.19. i) does not exceed

A^(BrQ A^(BR) kr^ ⁺ ~TR~

However, if k ^ o-, then

7,(BR) ^ ^(BR) _i_ ^(Br,) i __ ^ (Br,)

^ — R^ R-^— r? r ^ ~ rf On the other hand, if k ^- a, then

HBR) ^ -^(BR) _i_ HBr,) i ^ /(Br,)

^ — R^ J?/-(7— r? ri-^" ri Thus,

^^^maxf7^0 ^^2^ _^_^max^-^-,-^-^

and (1.19.2) follows.

Let us now consider the case in which, for each p > o, lim r~~P 7 (r) == oo as r —^ oo. We define, for a~ >> o,

„ ( „ ^(BR) . ^(Br)}

J ? , = s u p j J ? : ^^ = m f - ^ ^

We have -Rcr > o since ?i is continuous. If ^ > o- and -R ^ J?^, we have

^(BJ?) ^ ^(BJ?) i ^ 7(^J?^) i _ ^(BR^) R^ ~ R^ R^— R^ ^/-( T~ J?5

It follows that Ry is a non-decreasing function of o". Further if R^Ra, then ^(R)/^^ HR^IR^ so that

^R)^/^y

^(J?.)-YJ?J '

and it follows that R^ is an unbounded function of o". If we now let . ^ j ^ > o : ^) = inf 7(5r) for some a>o\,

( -n r>o r )

it follows from (1.19.2) that { Z ( J ? ) : - R € ^ J is a uniformly balanced, complete set of remainders of Z. The last assertion of the theorem follows from lemma 1.20 which states that ^^{ R: R^Ro} for some positive number -Ro.

We next consider the case in which

lim inf r~~i1 \ (r) == o as r -> oo for some p > o.

Let p (/) denote the smallest positive integer for which this holds. Let ^ be any unbounded set of positive real numbers such that

,. 'k(BR)

^ 7 ^ - ° RC^

and such that for all R € cK,

^-in,;^-^!.

Since lim inf ^~^P{/)^(B^) == o as r —^ oo, there is at least one such set dL The hypothesis that log ^ (e^) is convex implies that we may take

<^.== { R: R > o }, since r-P^l^Br) must decrease to o at oo. We construct Z' as follows. Let GL> be a primitive p(^)-th root of unity, say GO == exp(27Ti7p(^)). Let GO-^'Z denote the sequence ! c^-^ z//;, n = i, 2, 3, . . . , and then let

/?(/.)-!

Z ' = \J C^'Z.

/=0

It is clear that n(r, Z') = p(7) n(r, Z). Further, we have

(

S(fi, r,; k : Z') = ^ ^ ) S(r,, r,; k : Z).

/=o /

Consequently, for k == i, 2,..., p(?i) — T , we have that 5'(fi, r.2; k: Z) = o since the sum in parentheses is o for such values of /c. From the above equation, it also follows that

\ S ^ r , ' , k : Zf) \ ^ p W \ S ( ^ ^ , ; k : Z ) \

for all positive integers k. To prove, then, that { Z ' ( R ) : J?e^ } is a complete set of uniformly ^-balanced remainders of Z', it is sufficient to prove that (1.19.2) holds for k^p(V) and for every -Re^. As we have seen above, (1.19.2) is trivial unless r^^R^r.^, in which case

|S(/•„r,;/c:Z(J^))i=lS(R,r.;/c:Z)|^A^)+A^rL)•

However, in this case k ^ p(^) and ri^ R, so that

A(BR) __ KBR) i ^(Bn) i ^ ^(Br,)

J^ ""' '~RP^~ R^-PC^) —— ' ^(A) ' jr-^-^(A) '~ ^(A)

Hence (1.19.2) holds, and the proof is complete.

2. The Fourier coefficients associated with a sequence.

We now present the sequence of so-called Fourier coefficients associated with a sequence Z of complex numbers, and study its properties. We will use it in § 5 to construct an entire function f whose zero set coincides with Z, and to determine some properties of entire and meromorphic functions whose growth is restricted. The reason for calling them

<( Fourier coefficients " will become apparent on comparing their defini- tion with lemma 4.2 of section 4.

2.1. DEFINITION. — We define, for k = i, 2, 3, ..., S ' ( r ; k : Z ) = ^ ^ (I-)'

\^n\^r

S ' ( r ; k : Z ) = ^ ^ (^

\^n\^r

2.2. PROPOSITION. — We have

^(r;/c:Z) ^^N(er,Z).

Proof. — It is clear that \ S ' ( r ' , k : Z ) ^n(r)/A:, and we also have

^ ) ^ f ^n(ftdt=N(e^).

J,. L

2.3. DEFINITION. — Let a = = { a / : j , / c = i , 2, 3, ..., be a sequence of complex numbers. The sequence { c^(r; Z : a) j, k = o, ± i, ± 2, ..., defined by

(2.3.i) Co (r; Z : a) == Co (r; Z) = N(r, Z), (2.3.2) c,(r; Z : a) = - i a,+ S(r; k : Z)^f

2

— ' ^ ( r ^ i Z ) for k=i, 2, 3, ..., (2.3.3) c,(r;Z:a)=(c,(r;Z; a)/ for k == i, 2, 3, ...,

where * denotes complex conjugation, is said to be a sequence of Fourier coefficients associated with Z.

2.4. DEFINITION. — A sequence {c/:(r; Z : a ) } of Fourier coefficients associated with Z is called A-admissible if there exist constants A, B such that

(2.4. i) [ c,(r : Z; a) | ^ ^^ (k = o, ± i, ± 2, ...).

2.5. PROPOSITION.— A sequence Z is ^-admissible if and only if there exists a ^-admissible sequence of Fourier coefficients associated with Z.

BULL. SOC. MATH. — T. 96, FASC. 1. 5

66 L. A. RUBEL AND B. A. TAYLOR.

Proof. — Suppose that Z is ^-admissible. Then by 1.16, Z is strongly /-poised. Let a = (^-), k== i, 2, 3, ..., be the relevant constants, and form { Ck(r; Z : a) j from them by means of (2.3. i)-(2.3.3). Now (2.4. i) holds for k = o and some constants A, B since Z has finite 7-density.

For k = ± i, ± 2, ±: 3, ..., we have

[ c,(r; Z : oc) | ^ ^ [ a,+ S(r; k) | + ^ [ ^(r; A) ].

Then an inequality of the form (2.4. i) holds by proposition 2.2 since Z has finite ^-density, and because Z is strongly ^i-poised with respect to the constants { o ^ j .

On the other hand, suppose that (2.4.i) holds. Then N(r)==c,(r)^At(Br\

so that Z has finite ^-density. Moreover,

^ (^ + S(r; ^)) = c,(r; Z : a) + ^ S1 (r; k) .AUBr) N(er) ^A^(eBr)

^-p^ri+-27r^—k— ^?

so that Z is strongly ^-poised. By proposition 1.16, it follows that Z is /-admissible.

2.6. PROPOSITION. — Suppose that Z and a={^} are such that Ck(r\ Z: a) \^A^(Br). Then { c^(r; Z : a) } is 7-admissible. In parti- cular, there exist constants A', jB', depending only on A, B, such that

[c^Z:^^^)

| ^ | + i Proof. — For k = i, 2, ..., we have

(2.6. i) | c,(r) ] ^ ^ [ a, + 5(r; ^) I + ^ | ^(r; /c) ] and

(2.6.2) ^ ] a,+ S(r; ^) | ^[ c,(r) [ + ^ | ^(r; /c) [.

Since Co(r)=N(r)^A?i(Br), Z has finite 7-density. Then by (2.2),

\ S ' ( r ; k)\ ^( i/A:) 0(^(0 (r))) uniformly for k==i, 2, 3, . . . , by which we mean that there are constants A^//, B" for which | S ' (r, k) \ ^ (i fk) A " ) (B" r).

From our hypothesis and (2.6.2), it then follows that

rk\^k+ S(r; k) \ = 0(^(0(r))) uniformly for k > o.

Then by proposition 1.13, we have that

r^^+S^; k)\^^00.{0(r))) uniformly f o r A - = i , 2 , 3 , ....

K.

Then, using (2.6. i), we have

^(r) | ^ - 0(^(0(r))) uniformly for k = i, 2, 3, . . . .

Since c_^(r) = (Ck(r))\ and since Z has finite upper /-density, the propo- sition follows immediately.

2.7. DEFINITION. — The quadratic semi-norm of a sequence [ Ck(r; Z : a ) } of Fourier coefficients associated with Z is given by

£ , ( r ; Z : o O = j ^ I c ^ Z : ^ ^ .

( ^ = - 0 0 )

2.8. PROPOSITION. — The Fourier coefficients { c/c(r; Z : a)} are A-admissible if and only if E.z (r; Z : a) ^ A t (Br) for some constants A, B.

Proof. — First, if

, / „ . , A^(B,r) ]c,(r;Z:a)[^-^A then £.2 (r; Z : a) ^ A / (Br), where B == Bi and

A=A - 2 (^y

On the other hand, suppose there are constants A, B for which E^(r\ Z : a)^A/(J3r). Then it is clear that i c/,(r; Z : a) ] ^At(Br) so that by proposition 2.6, { C k ( r ; Z : a)) is ^-admissible.

The next result will be used in § 5 to help develop the so-called genera- lized Hadamard product.

2.9. THEOREM. — Suppose that Z is ^-admissible. Then there exist a ^-admissible supersequence Z'^Z and a complete, uniformly ^-balanced set of remainders of Z¹, { Z ' (R) = Re ^ }, and a family {a(R) j, a(J?) == { ak(R) }, k == i, 2, 3, . . . , -Re cR, of sequences of complex numbers such that

(2.9.1) \c,(r;Zf(R):^R))\^A^I^ (k= o, ± i, ± 2, . . . ) for some constants A, B, and further that

(2.9.2) lim Ck(r; Z1 (R) : a (R)) == o

7 ? > 0 0

^€^1

for all r > o and k == o, ± i, ± 2 , .... If lim inf r-^ ? (r) == oo for all k > o, G5 r ->oo, fhen we may ZaJce Z7 == Z and ^l == { J?: J? ^ J?o! /'or 50/ne positive number Ro.

Proof. — Let Z', Z ' ( R ) , and dl be constructed as in the proof of theorem 1.19. Then for suitable constants A, B, we have that

n (r, Z') ^ A ^ (Br\ N(r, Z ' ) ^ A ^ (Br), I ^fr r • k • Z^I?^ 1 ^ ^^^^^ ^ A^Br2) l^^fi, r.,, K .z. W}\-==. —-^— + —~j^~k—

We now let a(R) = { a/:(J?)} be defined by equations (1.14. i) and (1.14.2) that occur in the proof of proposition 1.14. Then, as was proved there, we have that

| ^(R)^- S(r; k : Z'{R)) | ^ ^^l^- This inequality, together with 2.6. i, gives

| c,(r; Z^R): a(J?)) [ ^ ^^ + \ I ^(r; k: Z'(R)) |.

However, we have

[^(r;^:^^))!^^^;^^))^^^;^^'1^,

to that (2.9.i) holds for some (possibly different) constants A, B. To prove 2.9.2, it is enough to show that a^(J?)->o as -R—^oo through ^., since it is obvious that

lim 5(r; A:: Z'(fi)) = lim ^(r; /c: Z ' ( R ) ) = o.

7?^oo 7?^oo

^e^. ^e<^.

From the equations defining a^(J?) it is clear that a^ (J?)->o as R—^oo except possibly in the case where p(?i)<oo and A:^p(7). However, in this case, we have from (2.9. i) that, for all r < 7?,

-w ^'

and consequently that

2A^(BJ?)

\ock(R)\^——jy——•

However, the family (^ was constructed, in the proof of theorem 1.19, in such a way that for k^pQ), we have

,. ^(BR) lim ——r- = o,

T?^ oo Jrr

^€^1

so that

lim ock(R)== o.

7?>cc K€cn

The final assertions of the theorem follow from theorem 1.19, where the family (^ was constructed.

2.10. COROLLARY. — If the sequences {Ck(r; Z ' (R): a^R))} of Fourier coefficients are as described in theorem 2.9, then for each r > o,

lim £,(r; Z ' ( R ) :a(i?))=o.

7?>oo

^e^R.

Proof. — For each r > o and n = i, 2, 3, . . . , we have

lim sup [E,(r; Z ' ( R ) : a(J?))]2^ lim sup V | c,(r; Z ' ( R ) : a(J?))

R > 3 0 7?^30 ^"

J?>30 7?->-

^ € d l ^€(R. \k\>n^€dl ^€(R. l ^ - l > ^

^A^r))- 2 (jT^J-

\k\>n

The result follows on letting n tend to oo.

3. Sequences that are ^-balanceable.

In this section, were are concerned with the process of enlarging a sequence Z so that it becomes ^-balanced. Growth functions ?i for which this is always possible are called regular and give rise to associated fields of meromorphic functions with special properties. For example, see theorem 5.4. The principal results of this section are propositions 3.5 and 3.6 which give simple conditions that ^ be regular. In addition, we give in proposition 3.3 a simple characterization of ^-admissible sequences for the case ?i (r) = r?.

,3.1. DEFINITION. — The sequence Z is ^-balanceable if there exists a ^-admissible supersequence Z ' of Z.

3.2. DEFINITION. — The growth function ^ is regular if every sequence Z that has finite ^-density is ^-balanceable.

3.3. PROPOSITION. — Suppose that ^ ( r ) = r P where p > o. Then (i) the sequence Z is of finite ^-density if and only if lim sup r~Pn(r, Z) <<oo as r—^oo;

(ii) if p is not an integer, then every sequence of finite 7-density is /-admissible;

(iii) if p is an integer, then Z is 7-admissible if and only if Z is of finite

^-density and S(r; p : Z) is a bounded function of r;

(iv) the function / (r) = r? is regular.

Proof. — To prove (i), we have that n(r)=0(rP) whenever Z has finite /-density. On the other hand, if lim sup r~^n(r) <oo, then n(r)^Ar? for some positive constant A so that

N (r) == f ^-j n (0 dt^A p-1 r?-

^ o

To prove (ii), suppose that N(t)^AtP. Then so long as k 7^?, we have

<-••> .f><')^+^)(^^)-

For, on integrating by parts, we have that the integral is equal to

^_^+,r^,

n(r,)_n(r0 r 'n

^ H- [ J, t^1

r^ n J ^ t +

But

^(^r2) ^ 4 ^r? — ^^P ' ~ r r - ^ ~ H ?

and similarly

n(rO^A^(rQ rt — rf Moreover,

r

"w.^Ar'^a^—^'+i), w j, ^ A r

-,< ^

/

^ . '"i

/ ^ d ^ A / ^d^^^^^+

and the inequality (3.3. i) follows. Hence, so long as p is not an integer, every sequence Z of finite r?-density is rP-balanced.

To prove (iii), suppose that Z has finite rP-density and that p is an integer. Then by (3.3.i), we see that all the conditions that Z be

^-balanced are satisfied except for k = p. For this case, the condition that S(r,, r.2; p) be bounded by r^A ^(Br,) + r^A )(Br.,) for some A, B is precisely the condition that S(r; p) be bounded, as is quite easy to see.

To prove (iv), we observe first that if p is not an integer then ?. (r) = r?

is trivially regular by (ii). If p is an integer, and Z has finite rF-density, let Z' be the sequence obtained by adding to Z all numbers of the form co-'Z^, where GI)P=—i. Then Z' has finite rP-density and S(r; p ^ ^ ^ o for all r > o . Hence by (iii), Z' is /-P-admissible, and it follows that ^ (r) == r? is regular.

The next two results give simple conditions, both satisfied in case 7(r) = rP, that imply that \ is regular. We do not know whether there exists a growth function 7 that is not regular.

3.4. DEFINITION. — We say that the growth function ^ is slowly increasing to mean that ^(27*) ^M'h(r) for some constant M.

If }i is slowly increasing, it is easy. to show that for some positive number p, / (r) = 0(rP) as r—^oo.

3.5. PROPOSITION. — If ^ is slowly increasing, then ^ is regular.

3.6. PROPOSITION. — If log ^ (e^) is convex, then ^. is regular, The proofs of these results use the next lemma.

3.7. LEMMA. — The growth function ^ is slowly increasing if and only if there exist an integer po and constants A, B such that

/o n ^ r^CO^ ^A^B^) f

(3-7.i) j ^dt^--^ for p^po.

If /(r) ==. r?, then we may choose po = [pj + i.

Proof. — Suppose first that (3.7.i) holds. We may clearly suppose thatjB^i. Then

A^Br) r-HQ r^°^ ^

~pr^~-J, ^+lM-J^,^lu-p(2^^

whenever p^po. Taking p = p o , we have 7.({2B^)^M^(B^), where M:=A(2B)^,

so that /(r) is slowly increasing. Suppose next that l(r) is slowly increasing, say ^ (2 r) ^ M ?i (r). Then

r^-y ^^^(Q^^ H^^r) Hr)y. /My

j t^ ^dt -2j J^. IP^ ^dt —2j pr^ (2^-)^ — ^M prP Z^ \ 2p )

' k=0 ~ r k=0 k=0

Hence, if po is taken so large that 2^/?0> M, we have an inequality of the form (3.7.i). In case ^ (r) = rP, we have M = 2?, and the final assertion follows.

Proof of proposition 3.5. — Let ?i be slowly increasing, and let Z be a sequence of finite 7-density. Choose po as in the last lemma so that, f o r p ^ p o ,

r-Ut) AUBr) J, t^ — P^

Define

Po

Z' = [ j c^Z,

/l-=0

L. A. RUBEL AND B. A. TAYLOR.

where j7o=po+ i, ex) == exp (2 Tn/no), and w^Z = { w -/ lZ n } , n = I , 2, 3 , . . . . Then we have S(ri, r.^; k : Z ' ) = o for k == i, 2, . . . , po, since

i -j- ^ + ol)2/(: 4" • • • 4~ (y^0^ == o

so long as ^k^-1, and this is true for A: == i, 2, . . . , po. Hence, to prove that Z' is 7-balanced, we need consider only k > po. For such k, with r <.r', we have

/ T V I

^f ^dn(t,Z').

S(r,r';Jc:Z)

2 ^ -— \ ^y

I / - < 1 ^ ! < ^

On integrating by parts, we have

f ^dn(t,Z')n^, Z1) n^Z1) r- n(t,Z') {r'Y + rk +K^ t^1 dtf

Since Z is of finite ^.-density and n(r, Z/) = ( p o + I ) n(r, Z), we have n(r, Z^f)^Ai^(Bi^) for some constants Ai, Bi by proposition 1.9.

Since 7 is slowly increasing, we have ^(B^r) ^M^(r) for some positive constant A a > o. To complete the proof of the theorem, we have only to prove that

'n^Z^^^A'^B'r)

/

for some constants A', B ' . However,

r^^^^A.f^jd^^^^^

j^ ^-+1 — -j^ ^ — ^

since k > po.

Proof of proposition 3.6. — It is no loss of generality to suppose that r~P^(r)-><x) as r—^oo for each p > o, since otherwise ^ is slowly increa- sing by lemma 1.20., and then proposition 3.5 applies. Now for p == i, 2, 3, . . . , let Rp be the largest number such that

A(R,) . ,^(r)

——— = mf -—— 9

^ rp

and let jRo = o. Then, as was shown in the proof of theorem 1.19 with the numbers R^ we have that R^R^R^..., and that Rp->oc as p->oo. Further, by lemma 1.20, r~P^(r) decreases for r ^Rp and increases for r^Rp. We also have the inequality

(3.6.i) 2^(r)^2^(2r) if r^fi^_i,

FOURIER SERIES METHOD. 73 since by the above remark,

A(r)^ U^r) rp-i —(ar)^-1'

Now let Z be of finite ^-density. For convenience of notation, we suppose that N(r)^^(r) and n(r)^^(r), since we could otherwise replace the function ^(r) by the function A^(Br) for suitable constants A, B. We then claim that

(3.6..) jr'^o^^

i f J c ^ 2 ^ and r^r^Rp.

To prove (3.6.2), we first integrate by parts, replacing the integral by n(r') n(r) r'"n(0

W "r^ ' J, ^ Now

"(^ ^M^^^

(r')* -= (r'y = r'1

since r ^ r ' and /""^(r) is decreasing for r^Rp^Rk. Also,

r^dt^r^ ^I d^^^ ^I

^ j/^-^^J ^-j/^—p0-1^ rp k~=Zp^~p' since f-^ ^ (Q ^ r-^ ^ (r) for r^t^r^Rp. Thus

ridn(o^+^+^^.

^ ^W^ ^ + ^ ^ ^ — p ^

We have k/(k — p) ^ 2 since Jc ^ 2^, and (3.6.2) follows.

We now define Z' as follows. For each z^eZ with J?^_i< Zn\ ^Rp^

we introduce into Z ' the numbers

i i

zn9 w2119 • ' " c*^1^

where m = m(p) = 2^ and ex) := c»)(m) = exp (27ri/m). We make the following assertions :

(3.6.3) ^(r,Z/)—n(^_„ZO=2^(^(r)—n(J?^)) if R^^r^R,, (3.6.4) n(^,Z/)^2^(r) if r^Rp,

(3-6-5) N^Z^^^Hr) if r^R,, (3.6.6) f ^nftZ-)

r ¹ i

^k j ^^(O ^ A-^2^ and r ^r'^R^

(3.6.7) ^(r.r'^Z^o if r, r^T^ and k is not a multiple of ^\

The assertions (3.6.3) and (3.6.4) follow immediately from the definition of Z', while (3.6.5) follows from (3.6.4), and (3.6.6) follows easily from (3.6.3). To prove (3.6.7), it is enough to prove that S(r, r'; A:: Z⁷) = o if R^^r^r^R,, j ^p, and k is not a multiple of IP. But, in this case, we have

where

^(r.^^Z^y^^^Z),

y == i + ^k + GL)^ + . . . 4- oo(^-1)A',

where m = m ( j ) = 2 / and co= ^(/n) = exp (27:1/77?). Since /c is not a multiple of 2^, k is therefore certainly not a multiple of 2/, so that c^ i.

We then have

I —— (»)"-"

and our assertion is proved.

We now prove that Z' is ^-admissible. To see that Z' has finite

^-density, let r > o and let p be such that R^^r^R^ Then by (3.6.5) and (3.6.1), we have that N(r, Z')^ 2^(7-)^ 2^1 (27-).

To see that Z' is ^-balanced, let k be a positive integer and suppose that (XT^T-'. Write k in the form ^q, where ^ is odd. Then, by (3.6.7), ^ ( r . ^ A ^ Z ^ o if R^^<^f. Suppose r^J?^. Then S ( ^ , ^r; k : Zf) = S ( ^ , rf r; k : Zf) , where r^min (r^, ^), by (3.6.7).

However,

S(r,r";k:Z')\^^f'"^dn(t,Z').

By (3.6.6), this last term does not exceed /' idn(f),

and this, in turn, does not exceed 4^ ;>.(/•), by (3.6.2). Consequently we always have | S(r, r ' ; k: Z) ^4r-^,(r), so that Z ' is 7 balanced, and the proof is complete.

4. The Fourier coefficients associated with a meromorphic function.

4.1. In this section, we associate a Fourier series with a meromorphic function, and use it to study properties of the function. As we mentioned in the introduction, the results of this section are generalized versions of the results of the earlier paper [5], and the proofs are essentially the same. Our notation follows the notation of [5] and the usual notation from the theory of meromorphic and entire functions. We first recall the results from the theory of meromorphic functions that will be needed.

For a non-constant meromorphic function f, we denote by Z(f) [respec- tively W(/*)] the sequence of zeros (respectively poles) of /, each occurring the number of times indicated by its multiplicity. We suppose throughout this paper that f(o) ^ o, oc. It requires only minor modifications to treat the case where f(o) === o or f(o) === oo. By n (r, f) we denote the number of poles of f in the disc { z : \ z ^r 1. By N(r, f) we denote the function,

^.^flMl,,

•^0

and by m (r, f) the function

m(^f)=^^ log-IA^IdO,

where log^rc = max (log x, o). We have, of course, that n (r, f ) = = n (r, W(f)) and N(r, f) == N(r, W(f)). The Nevanlinna characteristic, that measures the growth of /, is the function

T ( r , 0 = m ( r , f ) + N ( r , f ) . Three fundamental facts about T(r, f) are that (4.1.2) r ( r , 0 = T ( r , ^ + l o g | f ( o ) | , (4.1.3) T(r,fg)^T(r,f)+T(r,g), (4.1.4) T(r, f + g) ^T(r, f) + T(r, g) + log2.

Proofs of these facts may be found in [2], pages 4 and 5. An easy conse- quence of (4.1.2) is that

(4.1.5) ^ f [log f(r^ d 6 ^ 2 T ( r , f ) + l o g f(o)\.

This follows from (4.1.2) by observing that the first term is equal to 7n(r, f) + m(r, i/f\ which is dominated by T(r, f) + T(r, i/f).

For the entire functions f, we use the notation M(r,f)=sup{\f(z) : z = r } .

The following inequality relates these two measures of the growth of f in case/" is entire :

(4.1.6) T(r, /•)^log-M(r, f)^^±^T(R, f)

for o ^ r ^ R. For a proof of this, see [2], page 18. We will use (4.1.6) mostly in the form

(4•l•7) T(r, f)^\og^M{r, f)^3T(^r, f), which results from setting R = ir in (4.1.6).

The following lemma, which is fundamental in our method, was proved in [1] and [5]. For completeness, we reproduce the proof of [5] here.

4.2. LEMMA. — If f(z) is meromorphic in \z ^R, with f(o) -^ o, oc, and Z(f)={zn}, W(f)=={Wn}, and if log (f(z))==^^ near z=o,

k=0

then for o <r^R, we have

(4.2.1) iog|/-(^9)[= ^ c,(r,/1)^, where the Ck(r, f) are given by

(4.2.2) Co(r,/-)=log[/-(o) + V l o g 7- - y l o g^

^^ | Zn A" Wk

\^n\^r \w,,\^r

= log I f(o) I + N ( r , - ' \ —N (r, f).

(4.2.3) For A = i , 2 , 3 , ...,

^•"—^A^sK^-^y]

-^ 2 [S)-(^)'}

(4.2.4) For / c = i , 2 , 3 , ....

c,(r,f)==c,(r,f))\

where * denotes complex conjugation.

There are appropriate modifications if f(o) = o or f(o) = oo.

Remark. — Observe that in the notation of § 1 and § 2, formula (4.2.3) becomes

(4.2.5) c,(r, f)== \ a^+ ^ ; S(r; k : Z(f))-S(r; k : W(f))}

2t Jti

-^{Sf(r;k:Z(f))-Sf(r',k:W(f)).

Proof. — We may suppose that f is holomorphic, since the result for meromorphic functions will then follow by writing f as the quotient of two holomorphic functions. We may further suppose that f has no zeros on { z : [ z \ == r } , since the general case follows from the continuity of both sides of (4.2.2) and (4.2.3) as functions of r. Formula (4.2.2) is of course, Jensen's Theorem, and (4.2.4) is trivial since log [ f\ is real.

To prove (4.2.3), write

h{r)^^f [log/^°)]cos(/c9)d9

for some determination of the logarithm, and A : = i , 2 , 3 , . . . . Then, by integrating by parts, we have

T /" ^ f (rp^\

h(r)=-^f ^^^sin(/c9)fre.ed0.

This may be rewritten as

j.^_ i r f'^[^ zp\

l k v ) -27: kij^ ^W)\^ -^ )

This last integral may be evaluated as a sum of residues, and on taking real parts, we get the A-th cosine coefficient of log| f\. Similarly, consi- dering the integral

W=^f[\og(f(re^]sm(kQ)d^

where the integration is now between 7T/2A: and 2 71+77/2 A", we get the A-th sine coefficient. On combining these, we get (4.2.3).

We now define the classes of functions that we shall study.

4.3. DEFINITION. — Let ^ be a growth function We say that f(z) is of finite ^-type, and write /'€ A, to mean that fis meromorphic and that T(r, f)^A^(Br) for some constants A, B and all positive r.

4.4. DEFINITION. — We denote by As the class of all entire functions of finite ^-type.

4.5. PROPOSITION. — Let f b e entire. Then fis of finite t-type if and only if\ogM(r, f)^A^(Br) for some constants A, B and all positive r.

Proof, — This follows immediately from (4.1.7).

Note in particular that if ^ ( r ) = r P then f e A if and only if f is of growth at most order p, exponential type. Note also that the definition given above coincides with the definition given in [5], since only slowly increasing functions (see definition 3.4) were considered there. A disad- vantage of the earlier definition was that the classes A contained only functions of finite order. A possible disadvantage of the present definition is that for functions ?i of very rapid growth [e.g. ^(r) == exp(exp(r))],

^(27*) is much larger than any constant multiple of ^.(r).

We also note that by inequalities (4.1.3) and (4.1.5), A is a field and A.E is an integral domain under the usual operations.

The main theorem of this paper is the following.

4.6. THEOREM. — Let f be a meromorphic function. If f is of finite

^-type, then Z(f) and W(f) have finite 7-density, and there exist constants A, B such that

(4.6.1) \c^f)\^A-)w ( ^ = o , ± i , ± 2 , . . . ) .

K | -f- I

In order that f should be of finite A-type, it is sufficient that Z(f) [or W(f}\

have finite ^-density, and that the weaker inequality (4.6.2) \Ck(r,f)\^AHBr)

hold for some (possibly different) constants A, B. Thus, in order that f should be of finite ^-type, it is necessary and sufficient that Z(f) have finite

^-density and that (4.6.i) should hold. It is also necessary and sufficient that Z(f) have finite A-density and that (4.6.2) should hold.

We will first give a proof based on the Fourier coefficients and later give an alternate proof.

Proof. — The order of the steps in the proof will be as follows. We first show that if f satisfies an inequality of the form (4.6.2), and if either Z(f) or W(f) has finite ^-density, then f must satisfy an inequality of the form (4.6. i). We then show that if f is of finite ^-type, then Z(f) and W(f) are of finite ^-density, and f satisfies an inequality of the form (4.6.2). Finally, we prove that if Z(f) [or W(f)] has finite ^-density and if f satisfies an inequality of the form (4.6.i), then f must be of finite ^-type.

We shall suppose that f(o)==i. The case f(o) = o or f(o) == oo causes no difficulty since we may multiply f by an appropriate power of z, and the resulting function will still be of finite /-type. This is because if liminf(?i(r)/logr) == o as r-> oo, then by a well known result, essentially Liouville's theorem, the class A contains only the constants.

Let us suppose that either Z(f) or W(f) is of finite /-density and that i0^, f)\== 0(^(0 (r))) uniformly for k=o, ±i, ± 2 , .... On consi-

dering the case k = o, we see that both Z(/) and W(f) have finite /-density. It is enough to prove that f satisfies an inequality of the form (4.6.1) for k== i, 2, 3, . . . . , since c-j, is the complex conjugate of Ck. We prove this exactly as proposition 2.6 was proved.

From (4.2.5), we have

(4.6.3) [ c,(r, f) ] ^ - | a,+ S(r; k : Z(f))—S(r; k : W(f)) \

2

+^\S'(r;k:Z(f)-)\+ ^\S'(r;k:W(f)) and

(4.6.4) ^ | a,- + S(r; k : Z(/))— S(r; k : W(f)) ^ | c,(r, /•) + ' | S'(r; k : Z(f)) + l1 S'(r; k : W(f))

.2 ^

Then, by proposition 2.2, for Z .= Z(f) or Z == W (f), we have [ ^(r; A : Z) | = k-^OO^O^r))) uniformly for k > o.

By (4.6.3), it is therefore enough to prove that

| a,+ S(r; k : Z(f))—S(r; k : W(ft) \ = k-^ r^ 0 (A(0 (r))) uniformly for k = i, 2, 3, . . . .

But, we already have from (4.6 ,,4) that

1 ak+ S(r; k: Z(f))—S(r', k: W(f)) ] == r-^0(^(0(r))) uniformly for such k.

Replacing r by r^A:1^/', and observing that r ' ^ a r , we have that

|a,+ S ( r ' , k : Z(f))—5(^; k : W(ft) = ^-^-^(/(O^))).

Thus, the assertion will be proved if we can show that, for Z=Z(f) and Z == W(f), we have

\S(^,^r;k:Z)\==k-i^-^0(^0(^))).

This was proved in proposition 1.11 (see 1.11. i).

Now suppose that f has finite /-type. Then N(r,W(0)=N(r,/')^T(r,f),

so that W (f) has finite /-density. By (4.1.2), the function iff also has finite }-type. Hence Z(f) == W(i//*) also has finite ^-density.

8o L. A. RUBEL AND B. A. TAYLOR.

To see that an inequality of the form (4.6.2) holds, note that

\Ck(r,f)\= -^f {log|/'(^⁹) }e-^dQ

^-^f_V°&\f^\ ^^r(r,o+iog f(o)\

by (4.1.5).

Finally, suppose that W(f) has finite ^-density and that (4.6. i) holds.

If Z(f) has finite ^-density, we apply the argument below to the func- tion i If. Then N(r, /) = 0(^(0 (r))). It remains to prove that

^ (^ f) = 0 (^ (0 (r))). However,

m ( ^ 9 f t^^^J[^{7 :}l^{l o g}l^{/ t}(^r^l ^de9 which by the Schwarz inequality does not exceed

(^J^J1^!^^9)!2^)1'2-

By ParsevaFs Theorem, we have, for suitable constants A, B,

^Jlog f(re^ ^6= ^ c.(r,0 ^A^Br^ J, {^-J' Hence m(r, f) = 0(^(0(r))), which completes the proof of the theorem.

Specializing theorem 4.6 to entire functions, we have the next result.

4.7. THEOREM. — Let f be an entire function. If f is of finite 7-type then there exist constants A, B such that

AUBr)

(4.7.i) \c,(r,f)\^-^ ( ^ = o , ± i , ± 2 , . . . ) .

It is sufficient, in order that f be of finite ^'type, that there exist (possibly different) constants A, B such that

(4.7.2) | c,(r, f) | ^A 7(Br) (k = o, ± i, ± 2, ...).

Thus, in order that f should be of finite ^-type, it is necessary and sufficient that (4.7.i) should hold, and it is also necessary and sufficient that (4.7.2) should hold).

Proof. — This result is an immediate corollary of theorem 4.6 since W (f) is empty in case fis entire.

We now give a direct proof of theorems 4.6 and 4.7, that does not depend on the formula for the Fourier series coefficients given in lemma 4.2, and that is independent of proposition 1.11. We wish to thank P. MALLIAVIN for suggesting that there might be a proof of this kind. It has also the advantage that it is immediately applicable to subharmonic functions. The steps are the same as in the above proof except that we must prove directly that if f is of finite ?i-type, then \c/,(r,f) ^([ k | + i)-'A^(Br) for suitable constants A, B.

Suppose that f is of finite ^-type. Then, by the Poisson-Jensen formula,

log | f(r^) = F(6) + G(6)—^(0), where, choosing ? > r, and writing z == re1^,

G(6)= ^ lo, ^P__P

^ W - Y log

P-

Z__Wn 0 -0

WnZ o'2

and

^(0) - — f P^ r, cp, 6) log | f(oe^ \ dcp.

" " ^ -r.

Here,

9 — r - P ( o , r , 9 , 9 ) = -

p'2—2rp cos(6 — ^ ) + r1

is the Poisson kernel. We choose p == 2 r. Then each term in the expres- sions for G(9) and Jf(6) is of the form

log w—a —log i—aw\,

where w == I e1^ and \a\^i. Using 4.2, we see that the Fourier

2

coefficients of such as function are, for k > o, either i . -x. (aY i , -,, (ay

— —;-(2ar— —- 2 / 0 ' / 2/c or

.-L f ^ Y - ^ . 2 k \ 2 a ) 2 k

BULL. SOC. MATH. — T. 96, FASC. 1.

L. A. RUBEL AND B. A. TAYLOR.

according S i S o < \ a \ ^Io rI< a \ ^ l . In either case, we see that

2 2

the A-th Fourier coefficient of G ( 6 ) — H ( Q ) cannot exceed

^ n ( 2 7 , n + n ( 2 r , ^ .

To estimate the A-th Fourier coefficient of F (9), we use the estimate [7]:

, , . I / 7T ^ \

Ck ^^( - F ,

where

, ( a , F ) = ^ j T \F(x+a)-F(x)\dx.

cx)j (a,

A simple estimate shows that

^ ( a , F ) ^ ^ ( a ) ^ f [log f(27^9) dO, where

c^(a) == sup P(2r, r, 9 + a, cp)—P(2r, r, 0, cp)

u; ?'

But it is easy to see that

GL>*(a)^i2 sup | cos(0 + a—?)—cos(9 — cp) j ^- c a,

^' ?

for a suitable constant c. Combining these estimates with the fact that f is of finite ^-type and proposition 1.9, we obtain the desired estimate

A^.(Br) Ck(r,f)^

The next results are proved from theorems 4.6 and 4.7 in much the same way that the corresponding results of [5] were proved from theorem 1 of that paper. For the sake of completeness, we include the proofs.

4.8. DEFINITION. -— For a meromorphic function /, we define

( T r^ , V

E^f)= — / log / - ( r ^ l ^ O .

t2 7^-^ )

Notice that if f is entire with f(o) = i, and if a = ; ^ ] is such that Ck(r, f) = Ck(r\ Z(f): oc), k = o, ± i, ± 2, . . . , then£,(r, f) = E,(r\ Z(f); a), where this last quantity is the one defined in definition 2.7.

4.9. THEOREM. — Let f be an entire function. If f is of finite l-type and i^q< oo, then

(4.9.i) E,(r,f)^AHBr) for suitable constants A, B and all r > o.

Conversely, if (4.9.i) holds for some q^i, then f is of finite t-type.

Proof. — If fis of finite ^-type, then by the Hausdorff-Young Theorem ([7], p. 190), the I// norm of log | f(re1^) [, as a function of 9, is bounded by the IP norm of the sequence { c / , j, where (i/p)+(i/g) == i. By theorem 4.7, this IP normals dominated by an expression of the form A'k(Br). Conversely, using Holder's inequality,

\^^\^^f^r\^log\f^)\\^

ivy

log I f(re^) | p9 ^A^(Br),

for suitable constants A, B, and it follows from theorem 4.7 that f has finite ^-type.

4.10. THEOREM. — Let f be a meromorphic function of finite ^-type, with f (0)^.0, oo. Then for each positive number z there exist positive constants a, (3 such that, for all r > o,

( 4•^{1 0}•^{I )} ^f^^Wr) ^log ^⁹) l l )^{r f 6}^^I+^£• Remark. — We have as a consequence that, for all r > o,

2^ J_ ^ IA^'QI^^^ ^{d 6} ^^{I + 8}'

which is somewhat surprising, even in case f is entire, since it is by no means evident that the integral is even finite.

Proof. — There is a number (3 > o such that

\c^f) ^_M_ - , _ ^ ^ . U^r) —\k\+i ^v °⁵ -^{1 9} -^{2 9} • • •^;- Let

Then

F(9)=F(0,r)=^log|/-(r^)].

F(9)=^y^, where ^=^7?

We may also suppose that the constant M satisfies

^FITO

by theorem 4.9. By a slight modification of [7] (p. 234, example 4), we know that for any such F there exists a constant a > o, where a depends only on M and s, such that

^jT e x p ( a | F ( 0 ) [ d 0 ^ i + 3 , from which (4.10. i) follows.

5. Applications to entire functions.

We present, in theorem 5.2, a simple necessary and sufficient condition on a sequence Z of complex numbers that it be the precise sequence of zeros of some entire function of finite /-type. The condition is that Z should be /-admissible in the sense of definition 1.15. This generalizes a well-known theorem of Lindelof (see the remarks following the proof of theorem 5.2). Our proof depends on theorem 4.7 and a method, presented in theorem 5.1, for constructing an entire function with certain properties from an appropriate sequence of Fourier coefficients associated with a sequence of complex numbers. In an appendix, we give an alter- nate proof of theorem 5.2, due to H. DELANGE. We also prove, in theorem 5.4, that ^ has the property that each meromorphic function of finite /-type is the quotient of two entire functions of finite ^-type if and only if A is regular in the sense of definition 3.2. Accordingly, propositions 3.5 and 3.6 give a large class of growth functions ^ for which this is the case, including the classical case 7 (r) == r?. Even this case seems not to be known.

Finally, we develop the so-called generalized canonical product, a detailed discussion of which is given in the remarks preceding the proof of lemma 5.5.

We turn now to our first task, the construction of an entire function f from a sequence Z and a sequence {c^(r; Z : a)} of Fourier coefficients associated with Z. We recall that we have assumed that Z = { Z n } is a sequence of non-zero complex numbers such that Zn-> oo as n-> oo.

5.1. THEOREM. — Suppose that { c / c ( r ) } == { Cx:(r; Z : a)}, k = o,

± i , ± 2 , . . . , is a sequence of Fourier coefficients associated with Z such that for each r> o, ^| c/,(r) ]² < oo. Then there exists a unique entire function fwithZ(f) = Z, f(o) =i, and c/,(r, f) = Ck(r) for k =o, ±i, ± 2, ....

Proof, — We define

<D(pe^)= ^ Ck^)e1^.

Since ^ [^(p)^ ^, this defines ^(pe^) as an element of L^—TT, T:]

for each p > o, by the Riesz-Fischer Theorem. For p > o, we define the following functions :

( 5 - 1 - 0

^•'

>

^^=^-

(5.1.2) P^)=-Yl Bp(z;zn),

(5.1.3) K(w•,z)=^±z^

(5.1.4) (?o(2) = exp { ^ J' K(w, z) <? («;) ^ j, (5.1.5) /•p(^==Pp(z)eF^).

We make the following assertions :

(5.1.6) The function fp is holomorphic in the disc ; z : \ z ] < p { and its zeros there are those Zn. in Z that lie in this disc.

(5.1.7) MO)=I-

(5.1.8) Ifr<p,then a(r, f) =c,(r).

Now (5.1.6) is clear from the definition of fp. Also, f, (o) = Pp (o) (?p (o) = (?p (o) f) 1^.

1 - - » 1 ^ 5 '

However,

^(o)=exp{^J ^^)^j=exp{c.(p))=n ^,

l ( ( ] - p l ^ l ^ P

and it follows that /'p(o) = i.

To prove (5.1.8), we see by lemma 4.2 that it is enough to show that, near z == o,

^gf^(z)==^^, k == i