A Multidimensional Version of Levin's Secular Constant Theorem and its Applications
We study holomorphic almost periodic functions on a tube domain with the spectrum in a cone. We extend to this case Levin's theorem on a connection between the Jessen function, secular constant, and the Phragmen-Lindeloof indicator. Then we obtain a multidimensional version of Picard's the...
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irk-123456789-76132010-04-07T12:00:39Z A Multidimensional Version of Levin's Secular Constant Theorem and its Applications Favorov, S.Yu. Girya, N. We study holomorphic almost periodic functions on a tube domain with the spectrum in a cone. We extend to this case Levin's theorem on a connection between the Jessen function, secular constant, and the Phragmen-Lindeloof indicator. Then we obtain a multidimensional version of Picard's theorem on exceptional values for our class. Розглянуто голоморфні майже періодичні функції в трубчастій області з конусом в основі. На такі функції розповсюджується теорема Б. Я. Левіна про зв'язок між функцією Йессена та індикатором Фрагмена - Ліндельофа. Як наслідок, для розглянутого класу функцій одержано деякий аналог теореми Пікара. 2007 Article A multidimensional version of Levin's secular constant theorem and its applications / S.Yu. Favorov, N. Girya // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 3. — С. 365-377. — Бібліогр.: 14 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/7613 en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study holomorphic almost periodic functions on a tube domain with the spectrum in a cone. We extend to this case Levin's theorem on a connection between the Jessen function, secular constant, and the Phragmen-Lindeloof indicator. Then we obtain a multidimensional version of Picard's theorem on exceptional values for our class. |
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Article |
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Favorov, S.Yu. Girya, N. |
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Favorov, S.Yu. Girya, N. A Multidimensional Version of Levin's Secular Constant Theorem and its Applications |
| author_facet |
Favorov, S.Yu. Girya, N. |
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Favorov, S.Yu. |
| title |
A Multidimensional Version of Levin's Secular Constant Theorem and its Applications |
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A Multidimensional Version of Levin's Secular Constant Theorem and its Applications |
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A Multidimensional Version of Levin's Secular Constant Theorem and its Applications |
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A Multidimensional Version of Levin's Secular Constant Theorem and its Applications |
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A Multidimensional Version of Levin's Secular Constant Theorem and its Applications |
| title_sort |
multidimensional version of levin's secular constant theorem and its applications |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/7613 |
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A multidimensional version of Levin's secular constant theorem and its applications / S.Yu. Favorov, N. Girya // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 3. — С. 365-377. — Бібліогр.: 14 назв. — англ. |
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| first_indexed |
2025-07-02T10:25:41Z |
| last_indexed |
2025-07-02T10:25:41Z |
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1836530471892156416 |
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Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 3, pp. 365�377
A Multidimensional Version of Levin's Secular Constant
Theorem and its Applications
S.Yu. Favorov and N. Girya
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:favorov_s@mail.ru
girya@mail.ru
Received September 1, 2006
We study holomorphic almost periodic functions on a tube domain with
the spectrum in a cone. We extend to this case Levin's theorem on a con-
nection between the Jessen function, secular constant, and the Phragmen�
Lindel�of indicator. Then we obtain a multidimensional version of Picard's
theorem on exceptional values for our class.
Key words: Almost periodic function, mean motion, secular constant,
Picard's type theorems.
Mathematics Subject Classi�cation 2000: 42A75, 30B50.
An almost periodic function with the bounded from below spectrum has some
speci�c properties. Namely, it extends to the upper half-plane as a holomorphic
almost periodic function f of exponential type (H. Bohr [2]), then log jf j and the
mean value of log jf j over a horizontal line (the so-called the Jessen function) are
of the same growth along the imaginary positive semi-axis (B. Jessen, H. Torne-
have [7] and B.Ja. Levin [9]). The last result (together with the discovered by
Ph. Hartman [6], and B. Jessen, H. Tornehave [7] connection between the Jessen
function, mean motions of argf(z), and a distribution of zeros for holomorphic
almost periodic functions on a strip) shows the regularity of functions of this
important class.
In the end of the last century, L.I. Ronkin created the theory of holomorphic
almost periodic functions and mappings de�ned on the tube domains of multidi-
mensional complex space [11, 12, 14]. The Jessen function of several variables,
introduced by him, plays the main role in the value distribution theory for almost
periodic holomorphic mappings.
Here we continue studying the class of almost periodic functions on a tube do-
main with the spectrum in a cone done in [4] and [5]. Namely, we �nd a connection
between the asymptotic behavior of the Jessen function and the polar indicator.
c
S.Yu. Favorov and N. Girya, 2007
S.Yu. Favorov and N. Girya
Then we introduce a multidimensional analogue of the secular constant and study
its asymptotic behavior. Also, we obtain a multidimensional version of Picard's
theorem on exceptional values for our class.
Let us give a more detailed description of the subject.
Suppose f is a 2�-periodic function with the convergent Fourier series f(x) =P
n�n0
ane
inx; n0 � 0; an0 6= 0. Then f(z) =
P
n�n0
ane
inz, z = x + iy; is a natural
extension of f(x) to the upper half-plane C
+ . Clearly, f(z) is a holomorphic
function of exponential type jn0j without zeros in some half-plane y > y0 and
lim
y!+1
y�1 1
2�
�Z
��
log jf(x+ iy)jdx = lim
y!+1
y�1 log jf(iy)j = �n0:
In [2] and [7], these properties were generalized to almost periodic functions f
with bounded from below spectrum under the condition �0 = inf spf 2 spf . One
should only replace the mean value over the period by the Jessen function
Jf (y) = lim
S!1
(2S)�1
SZ
�S
log jf(x+ iy)jdx (1)
the number n0 by �0, and make use of the Phragmen�Lindel�of Principle (see
a footnote in the proof of Th. 1).
Note that the limit in (1) exists for every holomorphic almost periodic function
on a strip fz = x + iy : a < y < bg and the function Jf (y) is convex on (a; b).
Then for all y 2 (a; b), maybe except some countable set Ef , we have
J 0f (y) = �cf (y); (2)
where
cf (y) = lim
��!1
argf(
+ iy)� argf(� + iy)
� �
is the mean motion, or secular number, of the function f ; here argf(x + iy) is
a continuous branch of the argument of f on the line y = const. By the way,
equality (2) and the Argument principle imply that the number N(�S; S; y1; y2)
of zeros of the function f in the rectangle fjxj < S; y1 < y < y2g� has a density
lim
S!1
(2S)�1N(�S; S; y1; y2) = J 0f (y2)� J 0f (y1) (3)
for all y1; y2 62 Ef . It can also be proved that f has no zeros on a substrip
f� < y < �g if and only if Jf (y) is a linear function on the interval (�; �). In
this case,
f(z) = eicf z+g(z);
�Zeros should be counted with multiplicities.
366 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
A Multidimensional Version of Levin's Secular Constant Theorem...
where g(z) is almost periodic on the strip fz = x+ iy : x 2 R; � < y < �g.
Thus, an almost periodic function f with the property �1 < �0 = inf spf 2
spf is extended to C
+ as a holomorphic almost periodic function. Then we get
��0 = lim
y!+1
log jf(iy)j
y
= lim
y!+1
Jf (y)
y
= lim
y!+1
J 0
f
(y) = � lim
y!+1
cf (y) (4)
(see, for example, [7, 10]).
In the case �0 62 spf , the function is also extended to C + as a holomorphic
almost periodic function; the equalities (4) are also valid, but the proof of the
second equality is complicated, and this is the contents of Levin's Secular Constant
Theorem [9, 10].
Note that there exists a natural connection between the distribution of ze-
ros of an almost periodic holomorphic function on the upper half-plane and the
con�guration of its spectrum:
Theorem B ([1]). Suppose that the spectrum spf of an almost periodic func-
tion f on C
+ is bounded from below. Then:
1) if �0 = inf spf � 0, then f(z) tends to a �nite limit as y ! 1 on C
+
uniformly in x 2 R;
2) if �0 = inf spf < 0 and �0 2 spf , then f(z) ! 1 as y ! 1 on C +
uniformly in x 2 R;
3) if �0 = inf spf < 0 and �0 62 spf , then the function f(z) takes every
complex value on the half-plane y > q � 0 for each q <1.
To discuss the multidimensional case, we need the following de�nitions.
Let z = (z1; : : : ; zn) 2 C p , z = x + iy 2 C p , x 2 Rp ; y 2 Rp . By hx; yi or
hz; wi denote the scalar product (or the Hermitian scalar product for z; w 2 C p).
By j:j denote the Euclidean norm on Rp or C p . Also, for x = (x1; x2; : : : ; xp) put
0x = (x2; : : : ; xp). Further, by TK denote a tube set
TK = fz = x+ iy 2 C
p : x 2 R
p ; y 2 Kg;
where K � Rp is the base of the tube set.
A vector � 2 Rp is called an "-almost period of the function f(z) on TK if
sup
z2TK
jf(z + �)� f(z)j < ":
The function f is called almost periodic on TK if for every " > 0 there exists
L = L(") such that every p-dimensional cube in Rp with the side of length L
contains at least one "-almost period of f . In particular, when K = f0g, we get
the de�nition of an almost periodic function on Rp .
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 367
S.Yu. Favorov and N. Girya
A function f(z), z 2 T
, where
is a domain in R
p , is called almost periodic if
its restriction to TK is an almost periodic function for every compact set K �
.
The spectrum spf of an almost periodic function f(z) on TK is the set of
vectors � 2 R
p such that the Fourier coe�cient
a�(y; f) = lim
S!1
1
(2S)p
Z
jxjj<S;j=1::p
f(x+ iy)e�ihx;�idmp(x) (5)
does not vanish on K; here mp is the Lebesgue measure on R
p . The spectrum of
every almost periodic function f is at most countable, therefore we have
f(x+ iy) �
X
an(y)e
ihx;�ni;
where f�ngn2N = spf and an(y) = a�n(y; f). Note that for any given countable
set f�ng the functionP
n2N n
�2eihx;�
ni is almost periodic on Rp with the spectrum
f�ng.
In [11] L.I. Ronkin introduced the notion of the Jessen function of an almost
periodic holomorphic function f on T
by the formula
Jf (y) = lim
S!1
1
(2S)p
Z
[�S;S]p
log jf(x+ iy)jdmp(t):
Using the methods of the theory of distributions and peculiar properties of zero
sets for holomorphic functions, L.I. Ronkin con�rmed that the limit exists and
de�nes a convex function in y 2
. He also established the multidimensional
analogue of equality (3)
lim
S!1
m2p�2fz = x+ iy : x 2 [�S; S]p; y 2 !; f(z) = 0g
(2S)p
= �p�J(!);
where �J is the Riesz measure of J(y), ! � ! �
, �J(@!) = 0, and the area of
zero sets is taken counting the multiplicity.
Also, in [13] L.I. Ronkin proved that the products bn(y) = an(y)e
hy;�ni do
not depend on y for every holomorphic almost periodic function f(z) on T
; in
particular, the coe�cient b0 corresponding to the exponent � = 0 does not depend
on y. In the case, the Fourier series turns into the Dirichlet series
f(z) �
X
�n2Rp
bne
ihz;�ni; bn 2 C : (6)
In [12] L.I. Ronkin obtained the following results.
368 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
A Multidimensional Version of Levin's Secular Constant Theorem...
Theorem R. Let f be a holomorphic almost periodic function on T
. Then
the function Jf (y) is linear on the domain
0 �
if and only if the function f
has no zeros in T
0 . Moreover, in this case
f(z) = expfihcf ; zi+ g(z)g; z 2 T
0 ; (7)
where cf 2 Rp and g(z) is an almost periodic function on T
0 .
In conditions of Th. R, we have
Jf (y) = �hcf ; yi+Re b0; y 2
0;
where b0 is the corresponding coe�cient of the Dirichlet-series expansion of the
function g. Therefore, the following de�nition seems to be natural.
De�nition. The function �gradJf (y); y 2
; is the secular vector of the
almost periodic holomorphic function f on T
.
In order to formulate our results, we need some de�nitions and notations.
A cone � � R
p is the set with the property y 2 �; t > 0 ) ty 2 �: We will
consider the convex cones with nonempty interior and such that �
T
(��) = f0g.
By b� denote the conjugate cone to �, i.e., b� = fx 2 Rp : hx; yi � 0 8y 2 �g; note
that
bb� = �. As usual, IntA is the interior of the set A, and HE(x) = sup�2Ehx; �i
is the support function of the set E � Rp .
Let f be a holomorphic almost periodic function on a tube T� with an open
cone � in the base. By de�nition, put
hf (y) = sup
x2Rp
lim
r!1
ln jf(x+ iry)j
r
; y 2 �:
The function hf is called the P -indicator of f (see [14, p. 245]).
Theorem A ([5]). Let � be a closed cone in Rp , and f(x) be an almost
periodic function on Rp . Then f is extended holomorphically to T
Intb�
with the
estimates
9b <1 8�0 = �0 � Intb� [ f0g 9B(�0) 8z 2 T�0 jf(z)j � B(�0)ebjyj; (8)
if and only if spf � � + � for some � 2 Rp . If this is the case, then f(z) is
almost periodic on T
Intb�
and for all y 2 Intb�
hf (y) = Hspf (�y): (9)
For almost periodic functions with bounded spectrum, equality (9) was proved
in [4].
The following theorem is the main result of our paper.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 369
S.Yu. Favorov and N. Girya
Theorem 1. Let � be a closed cone in R
p , and f(x) be an almost periodic
function on R
p such that f is extended holomorphically to T
Intb�
with estimates
(8). Then for all y 2 Intb�
lim
R!1
Jf (Ry)
R
= hf (y): (10)
Furthermore, the secular vector �gradJf (Ry) tends to gradHspf (�y) as R!1
in the sense of distributions.
R e m a r k. Since Jf (y) is a convex function, we see that the secular vector
is a locally integrable function on Intb�.
P r o o f. From the beginning assume that y0 = (1; 0; 0; : : : ; 0) 2 Intb�, and
we will prove (10) for y = y0.
Put F (z) = f(z)eihz; hf (y
0)y0i (supx2Rp jf(x)j)�1, u(z) = log jF (z)j. Note that
F (z) is an almost periodic holomorphic function on T
Intb�
and jF (x)j � 1 on
R
p . Applying the Phragmen�Lindel�of principle� on the complex one�dimensional
plane fx+ wy : w 2 C
+g, we get
u(x+ ity) � hF (y) t; 8 z = x+ iy 2 T
Intb�
; t > 0: (11)
Then
hF (y) = hf (y)� hy; hf (y0)y0i; hF (y
0) = 0: (12)
Take y = y0 in (11). We get
u(z1;
0 x) � 0 8 (x1;
0x) 2 R
p ; y1 � 0: (13)
Fix " > 0. Since supx2Rp limr!1 r�1u(x + iry0) = 0, we see that for some
x0 = x0(") 2 Rp , r = r(") > 0,
u(x0 + iry0) � �"r: (14)
Using the Poisson formula for the disc D(x01 + iR; R) = fz1 : jz1 � x01 �
iRj < Rg � C + with R > r, inequality (13), and Maximum principle for the
subharmonic function u(z1;
0 x0), we obtain
u(x01 + ir;0 x0)
� 1
2�
2�Z
0
u(x01 + iR+Rei ;0 x0)
R2 � (R� r)2
R2 � 2R(R � r)cos(�=2 + ) + (R� r)2
d
�Suppose g(z) is continuous on C+ , holomorphic on C
+ , and bounded on R function, which
satis�es the condition log+ jg(z)j = O(jzj) as jzj ! 1; then for z = x + iy 2 C
+ we have
jg(z)j � supx2Rjg(x)je
�+y, where �
+ = lim supy!+1 y
�1 log jg(iy)j (see [8, p. 28]).
370 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
A Multidimensional Version of Levin's Secular Constant Theorem...
� r
4�R
3�=4Z
�=4
u(x01+ iR+Rei ;0 x0)d � r(8R)�1 sup
2[�=4;3�=4]
u(x01+ iR+Rei ;0 x0):
Hence (14) implies that u(x01+iR+Rei 0 ;0 x0) � �8"R for some 0 2 [�=4; 3�=4].
The function u(z1;
0 x0) is subharmonic in z1 2 C
+ . Taking into account (13) and
the embeddings
D(x01 + 2iR; R) � D(x01 + iR+Rei 0 ; R+R =
p
2) � C
+ ;
we get
�8"R � 2
�R2(3 + 2
p
2)
Z
D(x0
1
+iR+Rei 0 ; R+R=
p
2)
u(z1;
0 x0)dm2(z1)
<
1
3�R2
Z
D(x0
1
+2iR;R)
u(z1;
0 x0)dm2(z1): (15)
Remind that this inequality is valid for all R > r.
Put uR(z) = R�1u(Rz). From (15) it follows thatZ
D(x0
1
=R+2i; 1)
uR(z1;
0 x0=R)dm2(z1) > �24�": (16)
Furthermore, Th. A implies that the function hf (y) is continuous. Since (12),
we get hF (y) < " for jy � y0j < pÆ with some Æ = Æ(") 2 (0; 1=(p + 2)). If we
replace in (11) y by y=jyj, x by Rx, and t by Rjyj, we obtain
uR(z) = R�1u(Rx+ iRy) � "jyj (17)
for all z from the tube domain
T Æ = fz = x+ iy : x 2 R
p ;
��y=jyj � y0
�� < pÆg:
By de�nition, put
A(x1) = D(x11 + 2i; 1)�D(x12; Æ) �D(x13; Æ) � � � � �D(x1p; Æ):
It can easily be checked that for all x1 = (x11;
0 x1) 2 Rp we have A(x1) � T Æ. Also,
we may assume that T Æ � T
Intb�
[ f0g. Then for all z1 2 D(x01=R + 2i; 1) the
function uR(z) is subharmonic in z2 2 D(x12; Æ); z3 2 D(x13; Æ); : : : ; zp 2 D(x1p; Æ).
Hence (16) implies Z
A(x0=R)
uR(z)dm2p(z) > �24Æ2p�2�p": (18)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 371
S.Yu. Favorov and N. Girya
Suppose that for some � 2 R
p we have
jF (x0 + � + iry0)� F (x0 + iry0)j � e�"r � e�2"r:
Then jF (x0 + � + iry0)j � e�2"r and u(x0 + � + iry0) � �2"r. Using the latter
inequality instead of (14), we obtain the relationZ
A(x0=R+�=R)
uR(z)dm2p(z) > �48Æ2p�2�p": (19)
Put u+
R
(z) = maxfuR(z); 0g; u�R(z) = maxf�uR(z); 0g. From (17) it follows
that for all x1 2 R
p and all z 2 A(x1) we have
uR(z) <
p
10": (20)
Therefore, by (19),Z
A(x0=R+�=R)
u�
R
(z)dm2p(z) =
Z
A(x0=R+�=R)
u+
R
(z)dm2p(z)
�
Z
A(x0=R+�=R)
uR(z)dm2p(z) � 52Æ2p�2�p": (21)
In the sequel we need the following lemma.
Lemma 1. Let g(x) be an almost periodic function in x 2 Rp . Then for any
� > 0 there exist a real L = L(�) and a set E = E1 � � � � �Ep, Ej 2 R, such that
Ej \ [a; a+ L] 6= ; for every a 2 R, j = 1; : : : ; p, and each � 2 E is an �-almost
period of g.
P r o o f. By Bochner's criterium�, any sequence tn 2 R has a subsequence
tn0 such that the functions g(x+ (tn0 ;
0 0)) converge uniformly in x 2 R
p . In other
words, the functions g(x1+ tn0 ;
0 x) converge uniformly in x1 2 R and 0x 2 Rp . By
Bochner's criterium, the function g(x1;
0 x) is almost periodic in x1 2 R uniformly
in 0x 2 Rp�1 . Hence there exist E1 2 R and L = L(�) such that E1\[a; a+L] 6= ;
for all a 2 R and
jg(x1 + t;0 x)� g(x;0 x)j < �=p 8x1 2 R; 80x 2 R
p�1 ; 8t 2 E1;
i.e., each � = (t;0 0) for t 2 E1 is an �=p-almost period of g(x). In the same way,
we �nd E2; : : : ; Ep. It is clear that every point of E1 � � � � � Ep is an �-almost
period of g.
�For almost periodic functions on R see [10, Ch. VI, �1], or [3, p. 14�16]; the proof for the
multidimensional case is similar.
372 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
A Multidimensional Version of Levin's Secular Constant Theorem...
Take S < 1, and let L = L("; r) be real from Lem. 1. It is not di�cult to
prove that if R > L
p
2, then there exist �11 ; : : : ; �
N1
1 2 E1, N1 � 2
p
2S + 2, such
that
N1[
m=1
�
x01 + �m1
R
�
p
2
2
;
x01 + �m1
R
+
p
2
2
�
� [�S; S]; (22)
and each point of [�S; S] is contained in at most two intervals. For the same
reasons, if R > L
p
2=Æ, then for j = 2; : : : ; p there exist �1
j
; : : : ; �
Nj
j
2 Ej ,
Nj � (2
p
2S + 2)=Æ, such that
Nj[
m=1
�
x0
j
+ �m
j
R
� Æ
p
2
2
;
x0
j
+ �m
j
R
+
Æ
p
2
2
�
� [�S; S]: (23)
Let F = f� = (�m1
1 ; : : : ; �
mp
p ) : 1 � m1 � N1; : : : ; 1 � mp � Npg. Note that
F contains at most (2
p
2S + 2)pÆ1�p elements. By de�nition, put
�(S; Æ) =
�
x+ iy : x 2 [�S; S]p; jy1 � 2j < 1p
2
; jyjj < Æp
2
; j = 2; : : : ; p
�
Combining (22) and (23), we get
[
�2F
A
�
x0 + �
R
�
� �(S; Æ): (24)
Applying Lem. 1 to the function F (x+ iry0) with � = e�"r�e�2"r and using (21)
for every � 2 F , we obtainZ
�(S; Æ)
u�
R
(z)dm2p(z) �
X
�2F
Z
A(x
0+�
R
)
u�
R
(z)dm2p(z) � 52(2
p
2S + 2)pÆp�1�p":
Therefore, we have
lim
S!1
1
(2S)p
Z
�(S; Æ)
uR(z)dm2p(z) � �52(
p
2�)pÆp�1": (25)
It follows from the de�nition of the Jessen function that
lim
S!1
1
(2S)p
Z
[�S;S]
uR(x+ iy)dmp(x) =
JF (Ry)
R
: (26)
The functions uR(z) are uniformly bounded from above for z 2 TÆ.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 373
S.Yu. Favorov and N. Girya
Applying the Fatou lemma to inequality (25), we getZ
jy1�2j<
p
2
2
;jy2j< Æp
2
;:::jypj< Æp
2
JF (Ry)dy � �52(
p
2�)pÆp�1"R; (27)
for all R > R(L; Æ; r; ").
To �nish the proof, we need the following simple lemma.
Lemma 2. Let g(t) be a convex negative function on [��; �]. Then g(0) �
��1
R
�
�� g(t)dt:
P r o o f. The assertion of Lem. 2 follows immediately from the inequality
g(t) � g(0)minf1� t=�; 1 + t=�g:
Note that (20) and (26) imply
JF (Ry) �
p
10"R (28)
for all y = (y1; : : : ; yp); jy1 � 2j < 1; jyj j < Æ; j = 2; : : : ; p: Further, the Jessen
function JF (Ry) is convex in y ([11]). Therefore the function
g(0y) =
Z
jy1�2j< 1p
2
JF (Ry)dy1 � 2
p
5R"
satis�es the conditions of Lem. 2 in each variable y2; : : : ; yp with � = Æ=
p
2.
Applying the lemma p� 1 times and using inequality (27), we obtainZ
jy1�2j< 1p
2
JF (Ry1;
0 0)dy1 � �40(2�)pR":
Since (13), we see that the integrand is negative. Moreover, it is convex, therefore
JF (Ry1;
0 0) is a monotonically decreasing function in y1. Then we have
JF ((2� 1=
p
2)Ry0) � �30(2�)pR":
The inequality is valid for all R > R(") and " > 0. Thus we have
lim
R!1
JF (Ry
0)
R
= 0: (29)
Since JF (y) = Jf (y)� hy; hf (y0)y0i, we obtain (10) for y = y0.
374 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
A Multidimensional Version of Levin's Secular Constant Theorem...
For an arbitrary y0 2 Intb� consider an orthogonal operator A : Rp ! R
p such
that A(y0) = y0. Put f1(z) = f(Az). Since hf1(y
0) = hf (y
0) and Jf1(y
0) = Jf (y
0),
we obtain (10) for y = y0.
Further, from (11) and Th. A it follows that the function Jf (Ry)=R is bounded
from above on every compact subset of Intb�. Then �x y1 2 Intb� and take s > 0
such that fy : jy � y1j � sg � Intb�. Whenever jy � y1j < s, we have
2Jf (Ry
1) � Jf (R(2y
1 � y)) + Jf (Ry)
and
Jf (Ry)
R
� 2 inf
R�1
����Jf (Ry1)R
����� sup
R�1
sup
jy�y1j�s
maxfJf (Ry); 0g
R
:
This means that the functions Jf (Ry)=R are uniformly bounded from below on
every compact subset of Intb�. Using (10) and the Lebesgue theorem, we obtainZ
Jf (Ry)
R
'(y)dmp(y)!
Z
hf (y)'(y)dmp(y) as R!1
for every test function ' on Intb�, i.e., (10) is valid in the sense of distributions as
well. Therefore,
gradJf (Ry)! gradhf (y) as R!1
in the sense of distributions and Th. A implies the last assertion of Th. 1.
Corollary 1. Suppose that all conditions of Th. 1 are ful�lled. If Hspf (y) is
nonlinear on (�b�), then f(z) has zeros on the set IntT
b�\fjyj>qg for each q <1.
P r o o f. Theorem A yields that the function hf (y) is nonlinear for y 2 Intb�.
Now Th. 1 implies that Jf (y) is nonlinear on the set fIntb� \ fjyj > qgg for each
q <1. Then Th. R implies that f(z) has zeros on IntT
b�\fjyj>qg.
Applications to distribution of values. Here we apply Th. 1 to prove the
multidimensional variant of Th. B:
Theorem 2. Let � � Rp be a closed convex cone and f(x) be an almost peri-
odic function on Rp that has a holomorphic extension f(z) to T
Intb�
with estimates
(8). Then:
1) if (spf n f0g) � �, then f(z) tends to a �nite limit as y ! 1; y 2 �0,
uniformly in x 2 Rp for all �0 = �
0 � Intb� [ f0g;
2) if (spf n f0g) � � + � with some � 2 spf \ (��) n f0g, then the function
f(z) tends to 1 as y ! 1; y 2 �0, uniformly in x 2 Rp for all �0 = �
0 �
Intb� [ f0g;
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 375
S.Yu. Favorov and N. Girya
3) if (spf n f0g) � � + � with some � 2 (spf n spf) \ (��) n f0g, then the
function f(z) takes every complex value on the set IntT
b�\fjyj>qg for each
q <1;
4) if (spf nf0g) � �+� with some � 2 spf n((��) [ �), then the function f(z)
takes every complex value, except for at most one, on the set IntT
b�\fjyj>qg
for each q <1;
5) if (spf n f0g) 6� � + � for all � 2 spf and spf 6� �, then the function f(z)
takes every complex value on the set IntT
b�\fjyj>qg for each q <1.
R e m a r k. It is clear that we can replace spf n f0g by spf in Cases 1�3.
Therefore Th. 2 gives, in a sense, a complete description of the value distributions
for our class of almost periodic functions.
P r o o f. Case 1 was proved in [4], Case 2 was proved in [5]. Reduce Case
3 to a one-dimensional one. Take y0 2 Intb� such that hy0; �ki 6= hy0; �mi for all
k 6= m, and put '(w) = f(wy0), w 2 C .
First, check that sp' = fhy0; �i : � 2 spfg. This is evident for �nite ex-
ponential sums. In the general case, take a sequence of Bochner�Feyer expo-
nential sums� Pn(x), which approximates f(x) on Rp. Since spPn � spf and
Pn(uy
0) ! '(u) uniformly on R, we see that sp' � fhy0; �i : � 2 spfg. On the
other hand, if � 2 spf , then
ahy0; �i(0; Pn(y
0u)) = a�(0; Pn)! a�(0; f) 6= 0 as n!1:
Therefore, ahy0; �i(0; ') 6= 0 and hy0; �i 2 sp'.
Note that hy0; �ni ! hy0; �i as �n ! �, �n 2 spf . Also, since y0 2 Intb� and
�� � 2 � for all � 2 spf , we get hy0; �i > hy0; �i. Therefore, inf sp' = hy0; �i
and hy0; �i 62 sp'. From Th. B, i. 3 it follows that f(z) takes every complex
value on the set fz = wy0 : Imw > qg for each q <1.
Let us consider Case 4. Let b0 be the coe�cient of series (6) corresponding
to the exponent � = 0. Then for any A 2 C n fb0g each function f(z) � A has
the spectrum spf [ f0g. Suppose that the support function Hspf[f0g(y) is linear
on (�b�). Then it is not di�cult to prove (for example, see [5, Lem. 2]) that
spf [ f0g � �0 + � with some �0 2 (��) \ (spf [ f0g). But this is impossible
in our case. Hence, the function Hspf[f0g(y) is nonlinear on (�b�): Now Cor. 1
yields that the function f(z)�A has zeros on IntT
b�\fjyj>qg for each q <1.
Let us consider Case 5. Let b0 be the same as in Case 4. The function
f(z)� b0 has the spectrum spf n f0g. Note that the support function Hspfnf0g(y)
�For almost periodic functions on R see [10, Ch.VI, �1], or [3, p. 38�45]; consideration in the
multidimensional case is similar.
376 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3
A Multidimensional Version of Levin's Secular Constant Theorem...
is nonlinear on (�b�). Hence Cor. 1 implies that the function f(z)� b0 has zeros
on IntT
b�\fjyj>qg for each q < 1. Further, for any A 2 C n fb0g the function
f(z)�A has the spectrum spf [f0g. If the support function Hspf[f0g(y) is linear
on (�b�), then spf [ f0g � �0 + � with some �0 2 (��) \ (spf [ f0g). The both
cases �0 = 0 and �0 6= 0 contradict to the conditions of Case 5. Therefore the
function f(z)�A has zeros on IntT
b�\fjyj>qg for each q <1.
References
[1] A.S. Besicovitch, Almost Periodic Functions. Cambridge Univ. Press, Cambridge
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[2] H. Bohr, Zur Theorie der Fastperiodischen Funktionen, III Teil. Dirichletentwick-
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[5] S.Yu. Favorov and N.P. Girya, The Asymptotic Properties of Almost Periodic
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