B ULLETIN DE LA S. M. F.
L.A. RUBEL
B.A. TAYLOR
A Fourier series method for meromorphic
and entire functions
Bulletin de la S. M. F., tome 96 (1968), p. 53-96
<http://www.numdam.org/item?id=BSMF_1968__96__53_0>
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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.
Introduction..........................................................
1. An analysis of sequences of complex n u m b e r s . . . . . . . . . . . . . . . . . . . . . . .
1.1. Definition : n (r, Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Definition : N (r, Z). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.
54
56
57
57
Proposition : N(r, Z) = V log(r/ [ zj). . . . . . . . . . . . . . . . . . . . . . . . .
57
Proposition : n (r, Z) = r (dfdr) N (r, Z ) . . . . . . . . . . . . . . . . . . . . . . . . .
Definition : 5 ( r ; A - : Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Definition : S (r,, 7*3; A-: Z). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Definition : Growth f u n c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Definition : Finite A - d e n s i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proposition : n (r, Z ) ^ A A (5r). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Definition : ^ - b a l a n c e d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proposition : Finite density and A-balanced implies strongly
^-balanced................................................
1.12. Definition : ^ - p o i s e d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.13. Proposition : Finite ^-density and A-poised implies strongly ^-poised.
1.14. Proposition : A-balanced if and only if ^-poised. . . . . . . . . . . . . . . . .
1.15. Definition : ^ - a d m i s s i b l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.16. Propsition : Conditions for A-admissibility . . . . . . . . . . . . . . . . . . . . .
1.17. Definition : Remainder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.18. Definition : Complete set of r e m a i n d e r s . . . . . . . . . . . . . . . . . . . . . . . .
1.19. Theorem : Existence of complete set of remainders. . . . . . . . . . . . .
1.20. Lemma : Convex functions, j — 7 / . ^ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. The Fourier coefficients associated -with a s e q u e n c e . . . . . . . . . . . . . . . . . .
2.1. Definition : S^r; k : Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Proposition : | 5"(r; k : Z) [ ^ (A--1) N (er, Z ) . . . . . . . . . . . . . . . . . . . . .
2.3. Definition : c^(r; Z : a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Definition : ^-admissible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
57
57
67
58
58
58
1.4.
1.5.
1.6.
1.7.
1.8.
1.9.
1.10.
1.11.
58
5g
59
5g
60
60
61
61
61
61
65
65
65
65
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.
2.6.
2.7.
2.8.
Proposition : A-admissible if and only if there exist ^-admissible
sequences of Fourier coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proposition : Condition for X-admissibility of { c^r; Z : a) } . . . . . . . .
Definition : E^r; Z : a ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proposition : { c ^ ( r ; Z : a) } are ^-admissible if and only if
E,(r; Z
65
66
67
a)^ A^(Br).......................................
67
2.9. Theorem Existence of ^-balanced complete set of remainders...
2.10. Corollary lim Eg^; Z' (J?): a(2?)) == o . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Sequences that are A - b a l a n c e a b l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Definition ^ - b a l a n c e a b l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Definition Regular. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Proposition : A-admissibility in case ^(r) = r ? . . . . . . . . . . . . . . . . . . .
3.4. Definition Slowly increasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Proposition : Slowly increasing implies regular. . . . . . . . . . . . . . . . . . .
3.6. Proposition : log ^(e-") convex implies A is r e g u l a r . . . . . . . . . . . . . . .
3.7. Lemma : Condition that )v be slowly i n c r e a s i n g . . . . . . . . . . . . . . . . . .
4. The Fourier coefficients associated "with a meromorphic function.. . . .
4.1. The Nevanlinna characteristic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Lemma : Expressions for the Fourier c o e f f i c i e n t s . . . . . . . . . . . . . . . .
4.3. Definition : Finite > v - t y p e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Definition : A g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5. Proposition : Condition that an entire function be of finite A-type...
4.6. Theorem : Conditions on the Fourier coefficients that a meromorphic function be of finite ^ - t y p e . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7. Theorem : Conditions on the Fourier coefficients that an entire
function be of finite A - t y p e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8. Definition : E^ (r, f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9. Theorem : E^(r, f) ^ A).(Br). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
69
69
69
69
69
71
71
71
71
76
76
76
77
77
77
4.10. Theorem : / exp ^ i + £ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
5. Applications to entire f u n c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Theorem : Existence of entire functions with prescribed Fourier
coefficients..................................................
5.2. Theorem : Conditions that a sequence be the sequence of zeros of an
entire f u n c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Theorem : Conditions that a sequence be the sequence of zeros of a
meromorphic function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4. Theorem : A == A^/A^ if and only if A is regular. . . . . . . . . . . . . . . .
5.5. Lemma : log M(r, f) ^ 3 E^{i r, f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6. Lemma : lim g^ (z) = i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7. Theorem : Generalized Hadamard product. . . . . . . . . . . . . . . . . . . . . . .
Appendix : Delange's proof of theorem 5 . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
References...........................................................
78
80
82
83
84
87
88
90
91
91
92
93
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
FOURIER SERIES METHOD.
55
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 meromorphic 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 meromorphic 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 functions 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]
56
L. A. RUBEL AND B. A. TAYLOR.
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 distribution 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 function 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 decomposition of the subharmonic function. For numerous reasons, though,
we have preferred to keep this paper in the context of entire and meromorphic 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
FOURIER SERIES METHOD.
57
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
N(r,Z)==
J
^nd, Z),,
-^di.
Q
1.3. PROPOSITION. — We have
N(r,Z)= ^ ^g^T
^™
I^|
Proo/*. — Note that
2 log——=riog^)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 ^ r 7 "^ 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.lO.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^
H.iu.2^
A
S<r 7-2,
r ' /kc' .7\
^(^i) +
, A
^(BrQ
A(ri.,
Z )I [^
^ —^—
—^—
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.ll.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)
FOURIER SERIES METHOD.
5g
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,
(1.12.1)
|a,+^(r;^:Z)
A^CBr)
T
k
'
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 )\ (^)|rp > 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) <2 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
/S-p- 0 T
^CBP/)
For values of A-, then, such that k ^ p (/), we define
(1.14.2)
^=lim—5(p/;A:).
/-^ao
To show that the limit exists, we prove that the sequence i S ( o / ; k) ;,
j = i, 2, ..., is a Cauchy sequence. Let
^ = ^(P/; ^) — 5(p,,; A-) == 5(p,,, p;; A:).
We have
|I A
. i^-^^^P^) , A/(Bp;)
^•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
3
I^+^A)!^
^^.
—
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 AA(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;
FOURIER SERIES METHOD.
(ii)
(iii)
(iv)
(v)
Z
Z
Z
Z
is
is
is
is
61
strongly 7-balanced;
t-poised;
strongly 1-poised;
^-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
logA^) 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 nondecreasing 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^
62
L. A. RUBEL AND B. A. TAYLOR.
Proof. — Since u is convex and increasing, we may write
U (x)==u(o)+
(X) == U (0
rh(f)dt,
^n
where h is a non-negative and non-decreasing function.
then
If x == log r,
^) = exp ju(o) + F(h(t)-a) dt\,
{
^0
;
from which the first assertion follows immediately. To prove the second
7/r^
assertion, observe that if lim sup —v^/ << oo, then h(t) must be boundedsay h (0 ^ C. Then
-^—) = exp ; u(x + log 2)— u(:r) j^exp { 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.
iy. 2;
r - /c.
k • Zf7?^
^^^^ -t+ 2A
^ Br2 )
\I ^Sfr
(Ti, r.2..,
L, {n)) \I ^ —-j^—
—^—
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)
FOURIER SERIES METHOD.
63
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,
^^^maxf 7 ^ 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".
if R^Ra, then ^(R)/^^ HR^IR^ so that
Further
^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.
64
L. A. RUBEL AND B. A. TAYLOR.
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).
(
^(A)-l
S(fi, r,; k : Z') =
Further, we have
\
^ ^ ) 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)\^pW\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)
J^
i
^(Bn)
i
^ ^(Br,)
""' '~RP^~ R^-PC^) —— ' ^(A) ' jr-^-^(A) '~
Hence (1.19.2) holds, and the proof is complete.
^(A)
FOURIER SERIES METHOD.
65
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 definition with lemma 4.2 of section 4.
2.1. DEFINITION. — We define, for k = i, 2, 3, ...,
S'(r;k:Z)=^ ^
^ (^
(I-)'
\^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(ft
J,.
L
dt=N(e^).
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)
(2.3.2)
Co (r; Z : a) == Co (r; Z) = N(r, Z),
c,(r; Z : a) = - if a,+ S(r; k : Z)
2
—'^(r^iZ)
(2.3.3)
c,(r;Z:a)=(c,(r;Z; a)/
for
for
k=i, 2, 3, ...,
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)
^-p^ri+-27r^—k—?
.AUBr)
N(er)
^A^(eBr)
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 particular, 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.
FOURIER SERIES METHOD.
67
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 proposition 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:oO=j ^ Ic^Z:^^.
(^=-00
)
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
V k := — oo
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 generalized 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?>00
^€^1
68
L. A. RUBEL AND B. A. TAYLOR.
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.,,
r • Kk .z.
• Z^I?^
1 ^ —-^—
^^^^^ +
^A
^Br2)
l^^fi,
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
^€^1
Jrr
FOURIER SERIES METHOD.
69
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>30
J?>30
7?^30
7?->-
^"
^^€dl
€dl
^€(R.
\k\>n
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)=rP 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.
7°
L. A. RUBEL AND B. A. TAYLOR.
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
^
Hr^
n
[
r 'n
+1
J,
J ^ tt^
But
^ r2
( ) ^ 4 r? — ^^P ?
'~rr-^~ H
and similarly
n(rO^A^(rQ
rt — rf
Moreover,
-,< ^
/ ^ . '"i
"w.^Ar'^a^—^'+i),
r n^wdj,^^AA /r
/
(p
A
r
^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.
71
FOURIER SERIES METHOD.
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
^A^B^)
j r^CO^
^dt^--^
/o n ^
(3-7.i)
f
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
r-HQ+lM r^°^
^Br)
lu
A^Br)
~pr^~-J,
^
-J^,^
-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
j
r^-y
^^^(Q^^
H^^r)
dt
dt
t^
'
-2j J^.
k=0
~
IP^
Hr)y. /My
—2j pr^ (2^-)^ — M prP Z^ \ 2p )
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,
forp^po,
r-Ut)
J, t^
AUBr)
—
P^
Define
Po
Z' = [ j c^Z,
/l-=0
72
L. A. RUBEL AND B. A. TAYLOR.
where j7o=po+ i, ex) == exp (2 Tn/no), and w^Z = { w - / l Z 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
VI
2
^
-— \ ^y
S(r,r';Jc:Z)
^f ^dn(t,Z').
I /-<1^!<^
On integrating by parts, we have
f ^dn(t,Z')
n^, Z1)
n^Z1)
r- n(t,Z')
+
+K
{r'Y
rk
^
t^1 dtf
Since Z is of finite ^.-density and n(r, Z / ) = ( p o + I ) n(r, Z), we have
n(r, Zf)^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 Aa> o. To complete the proof of the theorem, we have only
to prove that
/
'n^Z^^^A'^B'r)
t^
—
kr^
for some constants A', B ' . However,
j^r^^^^A.f^jd^^^^^
^-+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 increasing 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 -——
rp 9
^
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')
W
n(r)
"r^ '
r'"n(0
J, ^
Now
"(^ ^M^^^1
(r')* -= (r'y = r'
since r^r' and /""^(r) is decreasing for r^Rp^Rk. Also,
r^dt^r^
^ j/^-^^J
I
d^^^
I
0 1
^-j/^—p - ^ 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
zn9
w2119
i
•'"
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?^))
(3.6.4)
n(^,Z/)^2^(r)
if
r^Rp,
if
R^^r^R,,
7
^
L. A. RUBEL AND B. A. TAYLOR.
3 6 5
( - - )
(3.6.6)
N^Z^^^Hr)
if
r^R,,
f ^nftZ-)
r 1 ^^(O
i
^k j
(3.6.7) ^(r.r'^Z^o
^ A-^2^
if
and
r ^r'^R^
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
^(r.^^Z^y^^^Z),
where
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 : Z f ) = S ( ^ , r f r ; k : Z f ) , 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.
FOURIER SERIES METHOD.
75
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) [respectively 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,^+log|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 consequence 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).
76
L. A. RUBEL AND B. A. TAYLOR.
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
^^
\^n\^r
7
| Zn
^
--ylog
A
"
\w,,\^r
Wk
= log I f(o) I + N ( r , - ' \ —N (r, f).
(4.2.3)
For A=i,2,3, ...,
^•"—^A^sK^-^y]
-^ 2 [S)-(^)'}
\^n\^r
(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.
FOURIER SERIES METHOD.
77
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
lkv)
f'^[^
zp\
-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, considering 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.
?8
L. A. RUBEL AND B. A. TAYLOR.
Proof, — This follows immediately from (4.1.7).
Note in particular that if ^(r)=rP 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 disadvantage 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
K | -f- I
(^=o,±i,±2,...).
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-
FOURIER SERIES METHOD.
79
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)) + l 1 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'^ar, 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 function i If. Then N(r, /) = 0(^(0 (r))). It remains to prove that
^ (^ f) = 0 (^ (0 (r))). However,
m(^9ft
^^^J[ 7 : l l o g l / t ( r ^l
de9
which by the Schwarz inequality does not exceed
(^J^J 1 ^!^^ 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
(4.7.i)
AUBr)
\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.
FOURIER SERIES METHOD.
81
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
PZ__Wn
0
^ W - Y log
-0
WnZ
o'2
and
^(0) - — f P^ r, cp, 6) log | f(oe^ \ dcp.
" " ^ -r.
Here,
9—r-
P(o,r,9,9)=- 2
p' —2rp cos(6 — ^ ) + r 1
is the Poisson kernel. We choose p == 2 r. Then each term in the expressions 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
-,, (aY
(ay
ii ., -x.
— —;-(2ar—
—2/0' /
2/c
or
.-L f ^ Y - ^ .
2k \2a )
BULL. SOC. MATH. — T. 96, FASC. 1.
2k
82
L. A. RUBEL AND B. A. TAYLOR.
according S i S o < \ a \ 2^ I o r2 I < a \ ^ l . In either case, we see that
the A-th Fourier coefficient of G ( 6 ) — H ( Q ) cannot exceed
^n(27,n+n(2r,^.
To estimate the A-th Fourier coefficient of F (9), we use the estimate [7]:
,
Ck
,
.I
/
^^(
7T
^\
-F ,
where
cx)j, (a,
(a,F)=^jT
\F(x+a)-F(x)\dx.
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
Ck(r,f)^
A^.(Br)
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^
E^f)= —
/
27
t
^-^
,
log / - ( r ^ l ^ O
V/q
.
)
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.
FOURIER SERIES METHOD.
83
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,
\^^\^^fr\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 10 I)
•
•
^f^^Wr)
log
^9)ll)rf6^I+£•
Remark. — We have as a consequence that, for all r > o,
2
^ J_ ^ IA^'QI^^^ d6 ^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
- , _ ^ ^
.
°5 - 1 9 - 2 9 • • • ; -
v
Let
F(9)=F(0,r)=^log|/-(r^)].
Then
F(9)=^y^,
where
^=^7?
^4
L. A. RUBEL AND B. A. TAYLOR.
We may also suppose that the constant M satisfies
^FITO
rfO^M.
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 alternate 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, ....
FOURIER SERIES METHOD.
85
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
B
^•'za>|=^n^^=^\ P —— -Z/2.Z
(5.1.2)
P^)=-Yl Bp(z;zn),
(5.1.3)
K(w•,z)=^±z^
(5.1.4)
(?o(2) = exp { ^ J'
(5.1.5)
K(w, z) <? («;) ^ j,
/•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)
(5.1.8)
MO)=IIfr<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
86
L. A. RUBEL AND B. A. TAYLOR.
where the a/, are such that a = ; a/: j and c^(r) = ^(r; Z : a). That is,
near z == o,
1
/-p^)
-/^- -.
fet-2^^/p00
<5-^)
1
yS:=i
We now make this computation. First, we have that
B
Zn '—p 2
'^ ^) _
2,,
(Zn——2)(p2——ZnZ)
B^(Z; Zn)
p 2 ——Z,^
Z,,——Z
^©--ia)^=i
^=i
Thus,
UJ z
^) =
P~(Z)
2^
"p
"^
p
^o(z)
v^
.
near z == 9
°
pp(z)
where
T
^.=2;
(p-}-2
^ •
l^f^p
i^i^p
rr
V
^^ \
Y
/
\
For k==o, i, 2, ..., we write w = p ^ ? and c^(p)^?==^^. Then by
the definition of c/:(p), we see that
^2o=N(p,Z)
and
" — — + ^ 2 ^ -^ i ( t --.'. 3 .-)l^j^P
Then
<l^)==N(p,Z)+2^'+^!
/:=!
=N(p,z)+^{^^+^p^^y'j,
so that
1
—.
—. ///
^
r^ // z)—
^w =——.
2 //"r
A // \\ r^
\\ ^W
2
^(w)K,.(w,
1
2^iJ^^-r
7^l
w
W
^|<rl=^-
e> '(U))
(w)
C>
-—
— ' d i,,v ,
27^l
2 7 ^ l ^|^|
(^
——
—^
^
^ | ^ l = F. ^
—
where
^(^,2)=^K(U;,2)=
2W
^"^^"(w—z)2
But
i
r
_j^_
.^^^^^(^_^^=^-l
^^K.^.^—— 2 ) 2
for
^=0,1,2.....
^,
A,
^,
. . .,
FOURIER SERIES METHOD.
and
i r
——: /
/ -1 vI 7———rr
i 2 dw =0
^^i^V^ (w—z)
for
8?
Jc = i, 2, 3, ....
Hence
OpOO - V y , .k-i
^ )-z
^
where
^-"—^sjaHl)'!Hence, near -z = o, we have
fp(z) _P',(z)
Q,(z) _^
W~P^)
W~^
'
and (5.1.9) is proved.
It next follows from (5.1.6)-(5.1.8) that
(5.1.10)
if p'> p, then /p/ is an analytic continuation of fp.
For if we define, for | z \ < p,
f
F(z)-^(z),
(/
W
then
Ck(r, F) = Ck(r, jfp/)—^(r, fp) = ^(r)-CA(r) = o,
for o ^ r < p , and therefore [ F ( 2 ) ] = i . On the other hand, F(o)==i
and it follows that F is the constant function i.
We now define the function f of theorem 5.1 by setting f(^)=fp( 2 )
if p > z |. It is clear that f is entire, and, by (5.1.6), that Z(f)=Z*
Also f(o)=i, and Ck(r, f) == Ck(r, fp) for p > r , so that c^(r, f)==Ck(r).
An argument analogous to that used in proving (5.1.10) proves that fis
unique, and the proof of the theorem is complete.
We now characterize the zero sets of entire functions of finite ^-type.
5.2. THEOREM. — A necessary and sufficient condition that the sequence Z
be the precise sequence of zeros of an entire function f of finite t-type is that Z
be A-admissible in the sense of definition 1.15, that is, that Z have finite
^-density and be t-balanced.
proof. — If Z == Z (f) for some f e A^, then by theorem 4.7, the sequence
{ c / c ( r , f ) } , is a 7-admissible sequence of Fourier coefficients associated
with Z, and thus Z is ^-admissible by proposition 2.5. Conversely,
suppose that Z is ^-admissible. Then by proposition 2.5 there exists
a ^-admissible sequence {c/c (r) { associated with Z. Then by theorem 5.1,
88
L. A. RUBEL AND B. A. TAYLOR.
there exists an entire function f with Z== Z(f) and ; c^(r, f)}== {c/, (r);.
Then by theorem 4.7 and the fact that { c / c ( r ) } is /-admissible, it follows
that f e As, and the proof is done.
Remark. — This theorem generalizes a well-known result of Lindelof[3],
which may be stated as follows.
THEOREM. — Let Z be a sequence of complex numbers, and let o > o
be given. If p is not an integer, then in order that there exist an entire
function of growth at most order p, finite type, it is necessary and sufficient
that there exist a constant A such that n(r, Z)^AP. If p is an integer,
it is necessary and sufficient that both this and the following condition be
satisfied for some constant B :
Sir
-——
^B.
\ -Z-/Z ,
This result follows immediately from theorem 5.2 and the characterization of rP-admissible sequences given in proposition 3.3. Our
result shows that, in general, the angular distribution of the sequence
of zeros of a function, and not only its density, is involved in an essential
way in determining the rate of growth of the function.
We turn now to the second problem of this section, that of determining
when A is the field of quotients of the ring A^. We first prove the
following result.
5.3. THEOREM. — In order that a sequence Z of complex numbers be
the precise sequence of zeros of a meromorphic function of finite 7-type, it is
necessary and sufficient that Z have finite ^-density.
Proof. — The necessity follows immediately from the fact that if f is
a meromorphic function, then N(r, f)^T(r, f). For the sufficiency,
we remark first that the method used in proving theorem 5.1 can be used
to construct suitable meromorphic functions. Indeed, suppose that
we are given two disjoint sequences Z, W of non-zero complex numbers
with no finite limit point, and constants y^, k==i, 2, 3, . . . , such that
the coefficients defined by
Co(r)==N(r,Z)—N(r,W),
c,(r)= rk- {y,+ S(r; k : Z)—S(r; k : W)}
— ^ {S'(r; k : Z)—Sf(r', k : W)
c^(r)==(c^r)Y
(k= i, 2, 3, ...),
( ^ = 1 , 2 , 3 , ...),
FOURIER SERIES METHOD.
89
satisfy ^. [ c/,(r) I'^oo for every r >o. Then by defining
D^z; w,,) JJ 5p(z;0
l^l^p
and
,__Pp(z)Qp(2)
fp(^
^pco
one can show, as in theorem 5.1, that the meromorphic function defined
/X 2 )^/'?^) for P > 1 ^ 1 has zero sequence Z, pole sequence W, and
Fourier coefficients {Ck(r) j. It is therefore enough to prove that given
a sequence Z of finite ^-density, there exists a disjoint sequence W of
finite ^-density, and constants y^, k==i, 2, 3, . . . , such that the c^(r)
satisfy Ck(r)\^A^(Br) for some constants A, ^ and all r > o. For
then, by the first part of the proof of theorem 4.6, the c/,(r) must satisfy
the stronger inequality
, / ., ^A'HB'r)
(r>o)?
l^l^-yeFFT
bv
for some constants A', B ' , so that the function f synthesized from the c/c(r)
must be of finite ^-type by theorem 4.6.
Supposing now that Z= { Z n } has finite ^-density, we define W= { W n }
by Wn=== Zn+ £n, n= i, 2, 3, ..., where the £n are small complex numbers
so chosen that \Wn\=\Zn , n==i, 2, 3, ..., all the numbers Wn and z/,
are different, and such that
2T-T<^Then N(r, W)==N(r, Z) so that W has finite /-density. Hence
\Sf(^;k:Z)\==k-iO(^0(^)))
and
IS^r;^: W)]^^-^^^^)))
for
/c=i, 2, 3, ....
We define
V
^
^ — Ikl y^
v )-)\zJ
L'T^—^f\w.,
J - VL
k^\\zj \wn) r
It remains to prove that
rk
\ y,+ S(r; k : Z ) — S ( r ; k : W ) |= 0(^(0(r)))
9°
L. A. RUBEL AND B. A. TAYLOR.
uniformly for k= i, a, 3, .... Now
r1
- I ^+S(r; k : Z ) — S ( r ; k : W ) \
rk
2
i v \( ^ ( L . ^
k 1
l^»l>r
rk
2
{zj
'
\wJ (
^
(z»yI y (0*.
A- 2j
(W^-Zn)*
'
'
(K,,,y-(z,,y
^ I
—— 2
/C
S
\Sn\>r
However, | (»„/'—(z,,)* :^/!'£n|z,,|*-1, so that we have
r*
-[y*+5(r;A:Z)—5(r;/c:W)|
2
^
y
— '> ^
£„]
1^+1
2 Tln-^<°>^
5.4. THEOREM. — The field A of all meromorphic functions of finite
t-type is the field of quotients of the rings A^ of all entire functions of finite
^-type if and aniy if ^ is regular in the sense of definition 3.2, that is, if and
only if every sequence of finite ^-density is 7-balanceable.
Proof. — First, suppose that ?i is regular and that /"€ A. Then Z(f)
has finite ^-density by theorem 5.13. There then exists a sequence
Z' 3 Z (f) such that Z is ^-admissible. [We may suppose, by the remarks
preceding the proof of theorem 4.6, that f(o) ^ o, oo]. Then by
theorem 5.2, there exists a function ^eA^ such that Z(g)==Z\ Since
we have then that Z(g)CZ(f), the function h= gifts entire. However,
T(r, h)^T(r, g) + T^r, ^ = T(r, g) + T(r, f) - log [ f(o)
by (4.1.2) and (4.1.3), so that heA^, and f== gfh is the desired representation.
Conversely, suppose that A==A^/A^. Let Z have finite ^-density.
Then by theorem 5.3, there exists a function /'eA with Z(f)===Z.
We write f= gfh with g, he As. Then Z(g) is 7-admissible, and
Z(g) 3Z (f)= Z, and we have proved that ?i is regular.
We turn now to the so-called (< generalized Hadamard product ",
which is somewhat analogous to the canonical product of Hadamard,
which represents an entire function of finite order [this is the case of the
growth function ^ (r) == rP] as the limit of a sequence of functions, namely
the partial products, each having as its zeros those zeros of f that lie
in discs with center at the origin. The sequence converges to f not only
uniformly on compact sets, but also in a way consistent with the growth
FOURIER SERIES METHOD.
91
of f. The fact that this sequence is usually written as a product is not
essential, at least for our purposes. In the case of more general growth
functions 7, there still exists a sequence or family of functions with
the desired properties, and we call this sequence the generalized Hadamard
product. For completely general growth functions 7, it seems to be
necessary first to multiply f by a suitable entire function.
In the case ?. (r) == rP, once the function f and the growth function 7.
are given, there is a canonical or natural choice of the product. If
lim inf r-^(r)= o for each k== i, 2, 3, . . . , then there is still a canonical
choice. In general, there is some leeway in the choice, that comes from
the leeway in the choice of the constants a/: that appear in the Fourier
coefficients associated with the partial products.
Before presenting these facts in the form of theorem 5.7, we require
some lemmas.
5.5. LEMMA. — Suppose that the function f is holomorphic in the disct
{ z : z \ < o } and that r< p/2. Then
logM(r,f)^3E,^r,f),
where £'2 is the quantity defined in definition 4.8.
Proof. — First since f is holomorphic, we have
T ( r , f ) = m ( r , f ) ^ ^ log | f(re^) \ dQ^E,(r,f)
from Schwarz's inequality. However, since f is holomorphic for [ z |< p,
we have by (4.1.6) that logM(r, f)^3T(2r, f).
5.6. LEMMA. — Let di be an unbounded set of positive numbers. Let
{ g n : R ^ d i } be a family of entire functions such that ^(o)==i, g p has
no zeros in the disc { z : z | < R}, and such that
(5.6.1)
|c,(r,^)|^
A7{(Br)
| K | -f- I
(5.6.2)
(k=o,±i,±2, ...,J?€^),
lim Ck (r, ga) == o,
fi>oc
-Re^
for each Jc= o, ± i, ±2, ... and each positive r. Then
lim gn(z)=i
L^^
R^a
uniformly on compacts sets.
Proof. — Since ^(o)==i, it is enough to prove that log|^(^)|^o
uniformly on compact sets as R->co through <^. Now
log I 9^) | ^logM(r, ^) ^ 9——^ T(p, gn)
92
L. A. RUBEL AND B. A. TAYLOR.
for any p>r, by (4.1.5). However, as was shown in the proof of
lemma 5.5, T(r, f)^E^(r, f) if fis holomorphic in [ z : \z ^r\. Hence
log gn(re^ ^^±^E,(o, gn).
Moreover, if -R>p, we may apply the same argument to i/^. Doing
this, and observing that E^(r, f)==E^_(r, iff), we have
log|^(^°)]|^^£,(p,^).
However, from (5.6.1) and (5.6.2), we have that
E,(r,gn)=f ^ |^-(p,^)|^
\k=-^
-> o
)
as R-^oo through <X, and the result is proved.
5.7. THEOREM. — Let f be a non-constant entire function of finite
A-type. Then there exist :
(a) a function AeA/7, h not identically o,
(b) an unbounded set cR. of positive numbers and a family {fn: R^di}
of entire functions fn of finite A-type,
(c) constants A, B,
such that
(i) the zeros of fn are the zeros of fh in the disc [ z : \z\^R\\
(ii) fhlf/f-^i as R->co through cK, uniformly on compact sets;
(iii) log M(r, F) ^A/(Br) if F is any of the functions f, h, fn, or fhlfn.
Moreover, if lim inf r-^'^ (r) = oo as r->co, for each k==i, 2, 3, . . . ,
then we may take h(z)=i for all z. If logy^e") 15 a convex function,
then we may take h(z)== i for all z, and ^=={R : R^Ro} for some J?o>o.
We call the family [ fn: R e ^ j a generalized Hadamard product associated with f.
Proof. — As before, we may suppose that f(o)==i. Let Z==Z(f),
and let Z', iK, and Ck(r, 7?)= ^(r; Z ' ( R ) : a(JT)) be as given in
theorem 2.9. Then by theorem 5.2, there is an entire function /'*€ \E
such that Z(/'*)=Z'. We may also suppose that f*(o)==i. Also,
by theorem 5.1, there are entire functions ^/?eA^ with ^(o)=i,
Ck(r, gn)==Ck(r, R), and Z(g^= Z^R). Then by lemma 5.6,
lim gn(z)=i as R->cc uniformly on compact sets. Also,
I ^ (r, gn) \ = \ Ck (r, R) \ == (| k \ + i)-^ o (/ (o (r)))
uniformly in k and jR.
FOURIER SERIES METHOD.
Let fr.==rig^
9^
Evidently (i) and (ii) hold for fn. Further,
Ck O1, fn) | = ! c/, (r, F) — ^ (^ ^0
= ( ! / c +I)- 1 0(^0(r))), uniformly in k and J?.
Hence, there are constants A, B such that
c,(r,F)^———A7(5r)
/cl+i
if F is any of the functions f, h, f*=fh, g p == fh^fn, or f/}=fhlgr,. Hence
or each of these functions F we have
£,(r, F)^A'^(Br\
where
AI=A
^\ 1 /-^
k\+i_
2W
Assertion (hi) then follows from lemma 5.5.
The final assertions of the theorem follow from the assertions of
theorem 2.9, except for the choice of h as the constant function 1.
However, if we take Zr = Z, then in the above argument we can
replace h(z) by i.
Appendix.
In this appendix, we give an alternate proof of theorem 5.2, that does
not depend on the Fourier series method, although it does use proposition 1.14. It was communicated to the authors by H. DELANGE,
and we are grateful to him for his permission to publish it here.
We first recall the following well-known lemma (see, for example
A. I. Markushevitch [4], p. 85).
LEMMA. — Let g be analytic in {z:\z\<R}, g(o)==o, (7(z)==^a^.
If Re[g(z)]^M for \z\<R, then for all k^i,
]aA- ^iMR-^
Now to prove the theorem, suppose first that ^eAjs-, and let
Z==Z(f)== { Z n } be the sequence of zeros of f, supposing without
loss of generality that f(o)= i. As usual, we let
M(r)==M(r,/-)==max{|^(z)|:lz|^rj.
We suppose that /'(z)==exp V,a^
in some neighborhood of z==o.
94
L. A. RUBEL AND B. A. TAYLOR.
As before, Jensen's Theorem shows that Z has finite /-density. Next
for r > o, let
/^-f^n—^^'n
Then fr is an entire function with f^(o)=i and f,.(z)^zo for z ^r.
For each r > o, there exists a function ^, analytic in z < r, such that
gr(o) = o and exp [^r(^)] = /'r(^) for z \ < r. Letting 7-0 be the smallest
modulus of the Zn, we see that for [ z \ < ro we have
k
9r(z)^[^+S(^;k:Z)]z
.
^k
i
But, for [ z [ = 2r, and therefore for [ z ^27-, we have [ fr(z) \^M (2r).
Consequently,
-ZM^(z)]^logM(27')
^ ^ <^
and the lemma yields
v /
i] y.k +
i So (r;
/ k7 : Z)
^\ |i ^
^ ——°^
2logM(2r)
,
which shows that Z is /-poised, so that by proposition 1.14, Z is
^-balanced.
Suppose now that we are given a sequence Z == } 2,,} that has finite
7-density and is ^-balanced, hence /-poised. We construct a function
/'eA^ such that Z==Z(/). There exist positive constants A and B
and a sequence { a/^ j of complex numbers such that
^ log——^AUBr)
•^—
'2/z
and
\ a k + S ( r ; k : Z ) \ •-
A7.(Br)
^
for k ^ i and r > o. Thus, for every positive r, the series
V[a^+ S(r; A": Z)]^ converges when |2| <r. Let to be the smallest
modulus of the Zn. For r<Fo, this series reduces to Va/:^, so that
V a/^ converges for | z | < z-o. We define functions g and F, for
1 z | < fo, by g (z) ==V a^.^ and F (z) = exp [^ (z)]. We see that F can
FOURIER SERIES METHOD.
9^
be continued to an entire function f, for we may define gr and fr in [ z \ < r
by
gr(z)=^l^+S(r;k:Z)}z^
and
/.(z) = exp [gr(z)}.
For ] 2 | < ro, we obviously have
^n^-i:)^)Since /^(z) 7^ o for [ z \ < r, we see that Z is the sequence of zeros of f
Now it is clear that for z\^=r,
g,,(z) ]^,| a,+ S ^ r ; k : Z) \ Tk^^A~k (^Br) (^^A^Br).
Therefore, for ] z \ = r,
^)1=M.)H^-^)
=]exp[(7,,(z)]| TT
-»- A
1^1^2r
i- 2
^^
^;exp[AA(2Br)]} '[J [ i + — — }
1^1^2r
L
I ^n
J
It follows that
logM(r,O^A^8r)+
V l o g [ i + —z— 1 "^
L
\ n _\
Since for | Zn ^ 2 r it follows that
3r
i+
71
^'/l
we have
2
1"§['+^]^ 2 log^^A^(3Br).
l^|^-2r
Combining results, we have that
log^r.^^A/^fir),
so that f € A.E, and the proof is complete.
96
L. A. RUBEL AND B. A. TAYLOR.
REFERENCES.
[1] EDREI (A.) and FUCHS (W. H. J.). — Meromorphic functions with several deficient
values, Trans. Amer. math. Soc., t. 93, igSg, p. 292-828.
[2] HAYMAN (W. K.). — Meromorphic functions. — Oxford, at the Clarendon Press, 1964
(Oxford mathematical Monographs).
[3] LINDELOF (E.). — Fonctions entieres d'ordre entier, Ann. scient. EC. Norm. Sup.,
t. 41, 1905, p. 369-895.
[4] MARKUSEVIC (A. I.). — Entire functions. — New York, American Elsevier publishing
Company, 1966.
[5] RUBEL (L. A.). — A Fourier series method for entire functions. Duke math. */.,
t. 30, 1963, p. 437-442.
[6] TAYLOR (B. A.). — Duality and entire functions. Thesis, University of Illinois, 1960.
[7] ZYGMUND (A.). — Trigonometrical series, First edition. — Warszawa, Seminar.
Matem. Univ. Warsz., 1935 (Monografie matematyczne, 5).
(Manuscrit recu Ie i3 juin 1967.)
Lee A. RUBEL,
Dept of Mathematics,
University of Illinois,
Urbana, Illinois (Etats-Unis);
B. A. TAYLOR,
Dept of Mathematics,
University of Illinois,
Urbana, Illinois (Etats-Unis).