Subharmonic almost periodic functions
We prove that almost periodicity in the sense of distributions coincides with almost periodicity with respect to Stepanov's metric for the class of subharmonic functions in a strip {z belongs C : a < Imz < b}. We also prove that Fourier coefficients of these functions are continuous funct...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2005
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irk-123456789-1065742016-10-01T03:02:09Z Subharmonic almost periodic functions Rakhnin, A.V. Favorov, S.Yu. We prove that almost periodicity in the sense of distributions coincides with almost periodicity with respect to Stepanov's metric for the class of subharmonic functions in a strip {z belongs C : a < Imz < b}. We also prove that Fourier coefficients of these functions are continuous functions in Imz. Further, if the logarithm of a subharmonic almost periodic function is a subharmonic function, then it is almost periodic. 2005 Article Subharmonic almost periodic functions / A.V. Rakhnin, S.Yu. Favorov // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 2. — С. 209-224. — Бібліогр.: 8 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106574 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We prove that almost periodicity in the sense of distributions coincides with almost periodicity with respect to Stepanov's metric for the class of subharmonic functions in a strip {z belongs C : a < Imz < b}. We also prove that Fourier coefficients of these functions are continuous functions in Imz. Further, if the logarithm of a subharmonic almost periodic function is a subharmonic function, then it is almost periodic. |
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Article |
| author |
Rakhnin, A.V. Favorov, S.Yu. |
| spellingShingle |
Rakhnin, A.V. Favorov, S.Yu. Subharmonic almost periodic functions Журнал математической физики, анализа, геометрии |
| author_facet |
Rakhnin, A.V. Favorov, S.Yu. |
| author_sort |
Rakhnin, A.V. |
| title |
Subharmonic almost periodic functions |
| title_short |
Subharmonic almost periodic functions |
| title_full |
Subharmonic almost periodic functions |
| title_fullStr |
Subharmonic almost periodic functions |
| title_full_unstemmed |
Subharmonic almost periodic functions |
| title_sort |
subharmonic almost periodic functions |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/106574 |
| citation_txt |
Subharmonic almost periodic functions / A.V. Rakhnin, S.Yu. Favorov // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 2. — С. 209-224. — Бібліогр.: 8 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT rakhninav subharmonicalmostperiodicfunctions AT favorovsyu subharmonicalmostperiodicfunctions |
| first_indexed |
2025-07-07T18:39:44Z |
| last_indexed |
2025-07-07T18:39:44Z |
| _version_ |
1837014539508383744 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2005, v. 1, No. 2, p. 209�224
Subharmonic almost periodic functions
A.V. Rakhnin and S.Yu. Favorov
Department of Mechanics and Mathematics, V.N. Karazin Kharkov National University
4 Svobody Sq., Kharkov, 61077, Ukraine
E-mail:rakhnin@univer.kharkov.ua
favorov@assa.kharkov.ua
Received January 21, 2005
We prove that almost periodicity in the sense of distributions coincides
with almost periodicity with respect to Stepanov's metric for the class of
subharmonic functions in a strip fz 2 C : a < Imz < bg. We also prove
that Fourier coe�cients of these functions are continuous functions in Imz.
Further, if the logarithm of a subharmonic almost periodic function is a sub-
harmonic function, then it is almost periodic.
Mathematics Subject Classi�cation 2000: 42A75, 30B50.
Key words: subharmonic functions, almost periodic functions in the sense of distri-
butions, Fourier coe�cients.
Subharmonic almost periodic functions were introduced in [2] in connection
with investigation of zero distribution of holomorphic almost periodic functions in
a strip. In this paper almost periodicity was de�ned in the sense of distributions,
namely as almost periodicity of the convolution with a test function. However,
subharmonic functions log jf(z)j, where f(z) is a holomorphic almost periodic
function, were considered much earlier in papers [5] and [6], where the important
point was to prove almost periodicity of such functions in the sense of distri-
butions. In [4] this was extended to a subharmonic uniformly almost periodic
function whose logarithm is a subharmonic function.
In this paper we prove that subharmonic almost periodic in the sense of distri-
butions functions are almost periodic in the classical sense, if we consider Stepanov
integral metric instead of the uniform metric. Therefore the classes of subhar-
monic almost periodic in the sense of distributions functions and subharmonic
Stepanov almost periodic functions are the same.
Now the Fourier�Bohr coe�cients of such functions can be de�ned in the
usual way. For a horizontal strip these coe�cients are functions depending on
Imz. In this paper we prove that these coe�cients depend continuously on Imz,
which allows us to approximate any subharmonic almost periodic function by
c
A.V. Rakhnin and S.Yu. Favorov, 2005
A.V. Rakhnin and S.Yu. Favorov
exponential sums with continuous coe�cients in Stepanov metric. Thus we prove
that subharmonic almost periodic functions are Stepanov almost periodic in the
sense of the de�nition in [8].
In [2] it was proved that exp(u), where u is a subharmonic almost periodic
in the sense of distributions function, is also almost periodic in the sense of dis-
tributions. Moreover, for an almost periodic function log jf(z)j, where f(z) is
a holomorphic function, jf(z)j is uniformly almost periodic. Conversely, we prove
that the logarithm of a subharmonic almost periodic function is an almost peri-
odic function, provided it is a subharmonic function. Thus we obtain a stronger
than the one in [4], as well as the converse to the result in [2].
We start with the following de�nitions and notations (see [1, p. 51]).
De�nition 1. A continuous function f(z) (z = x+ iy), de�ned on R + iK,
where K is a compact subset of R (it is allows that K = f0g), is called uniformly
almost periodic (Bohr almost periodic), if from any sequence ftng � R one can
choose a subsequence ftn0g such that the functions f(z + tn0) converge uniformly
on R + iK.
Equivalent de�nition is the following:
For any " > 0 there exists L(") > 0 such that each interval of length L(")
contains a real number � with the property
sup
z2R+iK
jf(z + �)� f(z)j < ":
De�nition 2. A distribution f(z) 2 D0(S) of order 0 (S is an open horizontal
strip) is called almost periodic, if for any test function ' 2 D(S) the convolution
Z
u(z)'(z � t)dxdy
is uniformly almost periodic on the real axis.
Note that according to [6], for an almost periodic distribution f(z) from any
sequence fhng � R one can choose a subsequence fhn0g, such that
R
'(z)f(z +
hn0)dxdy converge uniformly on every set �K = f'(z + t) : t 2 R; ' 2 Kg, where
K is a compact subset of D(S).
Any subharmonic function is locally integrable, so we can consider it as a dis-
tribution.
A class of subharmonic almost periodic functions in an open strip S will be
denoted by WAP (S).
Furthermore, for �1 < � < � < +1 we de�ne
S[�;�] = fz 2 C : � � Imz � �g;
210 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
Subharmonic almost periodic functions
ImS = fImz : z 2 Sg;
and for functions u, v, which are integrable on horizontal intervals in S[�;�], we
denote
d[�;�](u; v) := sup
z2S[�;�]
1Z
0
ju(z + t)� v(z + t)jdt:
De�nition 3. A function f(z) integrable on horizontal intervals in an
open horizontal strip S is called Stepanov almost periodic, if from any sequence
fhng � R one can choose a subsequence fhn0g and a function g(z) such that the
functions f(z + hn0) converge to g(z) in the topology de�ned by seminorms d[�;�],
�; � 2 ImS.
A class of a subharmonic Stepanov almost periodic functions in an open strip
S will be denoted by StAP (S). Since such functions are Stepanov almost periodic
on every line y = const, for u 2 StAP (S) there exists the mean value
M(u; y) := lim
T!1
1
2T
TZ
�T
u(x+ iy)dx:
To each such u we can associate Fourier�Bohr series
u(z) �
X
�2R
a�(u; y)e
i�x;
where
a�(u; y) :=M(ue�i�x; y):
are Fourier�Bohr coe�cients.
De�nition 4. A function u(z) � 0 is called logarithmic subharmonic in
a domain G � C , if the function log u(z) is subharmonic in this domain.
It is easy to see that a logarithmic subharmonic function is subharmonic.
We prove the following theorems:
Theorem 1. u(z) 2WAP (S) if and only if u(z) 2 StAP (S).
Theorem 2. Let u(z) be a logarithmic subharmonic function in a strip S.
Then log u(z) 2WAP (S) if and only if u(z) 2WAP (S).
Theorem 3. Let u(z) be a subharmonic almost periodic function in a strip S.
Then its Fourier�Bohr coe�cients are continuous in ImS.
From Theorem 3 and Bessel inequality for Fourier�Bohr coe�cients it follows
that spectrum of an almost periodic subharmonic function u(z) (i.e., the set
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 211
A.V. Rakhnin and S.Yu. Favorov
f� 2 R : a�(u; y) 6� 0g) it is most countable, which also follows from Theorem 1.12
in [6].
Theorem 4. Subharmonic function u(z) in an open horizontal strip S is
almost periodic if and only if there exists a sequence of �nite exponential sums
Pm(z) =
NmX
n=1
a(m)
n (y)ei�
(m)
n x; (1)
where �n 2 R, a
(m)
n (y) 2 C(ImS), which converges to the function u(z) in the
topology de�ned by seminorms d[�;�], �; � 2 ImS.
Moreover, Pm(z), m = 1; 2; : : : are subharmonic functions in S.
To prove the theorems above we use the following propositions:
Proposition 1. Convergence of subharmonic functions in D0(G) is equivalent
to the convergence in L1
loc
(G) (see [7]).
Proposition 2. Weak limit of subharmonic functions is subharmonic function
(see [7]).
We denote by G� the Green potential of a measure � for the disk B(R; 0), i.e.
G�(z) :=
Z
B(R;z0)
log
jR2 � z�j
Rjz � �j
d�(�):
Lemma 1. Let measures �n converge weakly to a measure � in a neighbor-
hood of the disk B(R; 0), and �(@B(R; 0)) = 0. Then for any t1 > 0, t2 > 0 such
that t21 + t22 < R2,
lim
n!1
sup
y2[�t2;t2]
t1Z
�t1
jG�n(z)�G�(z)j dx = 0; (2)
where z = x+ iy.
P r o o f. Denote �n = �n � �. We have
sup
y2[�t2;t2]
t1Z
�t1
jG�n(z)�G�(z)j dx � sup
y2[�t2;t2]
t1Z
�t1
�������
Z
B(R;0)
log
jR2 � z�j
R
d�n(�)
�������
dx
+ sup
y2[�t2;t2]
t1Z
�t1
�������
Z
B(R;0)
log jz � �jd�n(�)
�������
dx: (3)
212 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
Subharmonic almost periodic functions
The condition �(@B(R; 0)) = 0 implies that the restrictions of the measures
�n to the disk B(R; 0) converge weakly to the restriction of the measure � on
the disk, and the function log(jR2 � z�jR�1) is continuous for jxj � t1, jyj � t2,
� 2 B(R; 0). Thus the �rst term on the right-hand side of (3) is small. Without
loss of generality, we can assume that R < 1=2, so that for z; � 2 B(R; 0) we have
log jz � �j < 0.
Let " > 0 be an arbitrary �xed number. We denote log" jz��j = maxflog jz�
�j; log "g. This function is continuous for jxj � t1, jyj � t2, � 2 B(R; 0) and for
any " > 0. We have
sup
y2[�t2;t2]
t1Z
�t1
�������
Z
B(R;0)
log jz � �jd�n(�)
�������
dx � sup
y2[�t2;t2]
t1Z
�t1
�������
Z
B(R;0)
log" jz � �jd�n(�)
�������
dx
+ sup
y2[�t2;t2]
t1Z
�t1
Z
B(R;0)
jlog jz � �j � log" jz � �jj dj�nj(�)dx:
The �rst term on the right-hand side of this inequality is small when n is su�-
ciently large. Then
sup
y2[�t2;t2]
t1Z
�t1
Z
B(R;0)
jlog jz � �j � log" jz � �jj dj�nj(�)dx
= sup
y2[�t2;t2]
Z
B(R;0)
Z
[�t1;t1]\fx:jx+iy��j�"g
(log "� log jz � �j)dxdj�nj(�)
�
Z
B(R;0)
"Z
�"
(log "� log jxj)dxdj�nj(�) � 2"j�nj(B(R; 0)):
Note that since " > 0 is arbitrary, and measures �n weakly converge to zero, one
can choose a constant C 2 R with j�nj(B(R; 0)) < C. The lemma is proved.
Lemma 2. Let un(z) be a sequence of subharmonic functions in a domain
G � C converging to a function u0(z) 6� �1 in D0(G), and let
sup
z2G0
un(z) �W (G0) <1
for any subdomain G0 � G. Then for any rectangle [a; b] � [�;�] � G,
lim
n!1
sup
y2[�;�]
bZ
a
jun(z) � u0(z)j dx = 0: (4)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 213
A.V. Rakhnin and S.Yu. Favorov
P r o o f. For every disk B(z0; R) �� G we have the following representation
un(z) = �G�n
R
(z; z0) +HR(z; z0;un) n = 0; 1; 2 : : : ;
where �n are the Riesz measures of the functions un(z), G
�n
R
(z; z0) is the Green
potential of the measure �n in the disk B(z0; R), and HR(z; z0;un) are the best
harmonic majorants of the functions un(z) in this disk. Conditions of the lemma
imply that �n converge weakly to the measure �0. Without loss of generality, we
can assume that �(@B(z0; R)) = 0, and using Lemma 1 we conclude that for any
t1; t2, t
2
1 + t22 < R2
sup
y0�t2�y�y0+t2
t1+x0Z
�t1+x0
jG�n
R
(z; z0)�G
�0
R
(z; z0)jdx �! 0;
when n ! 1. From this it follows that the functions HR(z; z0;un) converge to
the function HR(z; z0;u0) in D
0(B(z0; R)). Now using the mean value property,
Harnak inequality, and obvious inequality
HR(z; z0;un) �W (B(z0; R)) <1; n = 0; 1; 2 : : : ;
we obtain uniform convergence of HR(z; z0;un) to the function HR(z; z0;u0) in
the rectangle [�t1 + x0; t1 + x0] � [�t2 + y0; t2 + y0]. Covering the rectangle
[�; �]� [a; b] by a �nite number of such rectangles, we prove the lemma.
P r o o f o f T h e o r e m 1. Inclusion StAP (S) � WAP (S) is obvious.
We prove the opposite inclusion. We consider arbitrary substrip S[�;�], �; � 2
ImS and a sequence fhjg � R. Since u(z) is a subharmonic almost periodic
distribution, there exists a subsequence fhjkg such that for some subharmonic
(clearly also almost periodic) function v(z) and for any ' 2 D(S[�;�]), uniformly
in t 2 R,
lim
k!1
Z
S
(u(z + hjk + t)� v(z + t))'(z)dxdy = 0: (5)
Now we will show that the functions u(z + hk) converge to v(z) in the topology
de�ned by seminorms d[�;�], �; � 2 ImS. Assuming the contrary, there exist
"0 > 0, �; � 2 ImS such that for an in�nite sequence k0
d[�;�](u(z + hj
k0
); v(z)) > "0;
and therefore there exists a subsequence ftk0g 2 R such that
sup
y2[�;�]
1Z
0
ju(z + hj
k0
+ tk0)� v(z + tk0)jdx > "0: (6)
214 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
Subharmonic almost periodic functions
Passing to a subsequence (if necessary), we can assume that
u(z + hj
k0
+ tk0)! w(z); v(z + tk0)! w1(z) in D
0(S[�;�]):
Lemma 2 implies
sup
y2[�;�]
1Z
0
ju(z + hj
k0
+ tk0)� w(z)jdx ! 0; k0 !1;
and
sup
y2[�;�]
1Z
0
jv(z + tk0)� w1(z)jdx! 0; k0 !1;
and thus inequality (6) implies
sup
y2[�;�]
1Z
0
jw(z) � w1(z)jdx � "0: (7)
On the other hand, using (5), for any test function '(z),
Z
S[�;�]
(w(z) � w1(z))'(z)dxdy = lim
k0!1
Z
S[�;�]
(u(z + hj
k0
+ t0k)� v(z + tk0))'(z)dxdy
= lim
k0!1
Z
S[�;�]
(u(z + hj
k0
)� v(z))'(z � tk0)dxdy = 0;
and thus w(z) = w1(z) almost everywhere. Since w(z) and w1(z) are subharmonic
functions, then w(z) � w1(z), which contradicts (7). The theorem is proved.
To prove Theorem 2 we need the following lemmas.
Lemma 3. Let '(t) be a function continuous in [�c; c]. Then for any " > 0
there exists Æ, depending on ' and ", such that for any two integrable on compact
set K functions f; g : K ! [�c; c] the inequality
Z
K
jf(x)� g(x)jdm < Æ
implies the inequality
Z
K
j'(f(x)) � '(g(x))jdm < ": (8)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 215
A.V. Rakhnin and S.Yu. Favorov
P r o o f. Choose � > 0 such that jt1� t2j < � implies j'(t1)�'(t2)j <
"
2m(K)
,
and denote
A1 = fx 2 K : jf(x)� g(x)j < �g;
A2 = fx 2 K : jf(x)� g(x)j � �g:
Notice that m(A2) �
1
�
R
A2
jf(x)� g(x)jdm, and therefore
Z
K
j'(f(x))�'(g(x))jdm �
Z
A1
j'(f(x))�'(g(x))jdm+
Z
A2
j'(f(x))�'(g(x))jdm
�
m(A1)"
2m(K)
+
2 sup j'(t)j
�
Z
K
jf(x)� g(x)jdm:
Choosing suitable Æ, (8) follows. The lemma is proved.
Lemma 4. Let un(z) be a sequence of uniformly bounded from above lo-
garithmic subharmonic functions in a domain G � C , converging to a function
u0(z) 6� 0 in the sense of distributions. Then the functions log un(z) converge to
the function log u0(z) in the sense of distributions.
P r o o f. The functions un(z) are logarithmic subharmonic, and in particular
subharmonic. Using Proposition 1, un(z) converge to u0(z) in L
1
loc
(G).
Next, these functions are uniformly bounded from above by some constant
V > 0, bounded from below by 0, and the function l"(t) = logmaxf"; tg is con-
tinuous in the interval [0; V ]. Lemma 3 implies that for �xed " the functions
l"(un)(z) converge to the function l"(u0)(z) in L
1
loc
(G), and thus in the sense of
distributions. From Proposition 2 it follows that the functions l"(u0)(z) are sub-
harmonic for all ", and their monotone limit when "! 0, i.e. the function log u0,
is also subharmonic.
Now we consider a disk B(z0; r) �� G. From the convergence in L1(B(z0; r))
of the sequence un(z) it follows that the subsequence fun0(z)g converges uniformly
on every �xed compact set K1 � B(z0; r) with positive Lebesgue measure. Since
the function log u0(z) is subharmonic and not identically �1 on K1,
sup
z2K1
(log u0(z)) � C0;
or
sup
z2K1
(u0(z)) � eC0 :
Thus for all n0 > n0
sup
z2K1
(un0(z)) � eC0�1;
216 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
Subharmonic almost periodic functions
and
sup
z2K1
(log un0(z)) � C0 � 1; 8n = 0; 1 : : : :
Since the functions un(z) are uniformly bounded from above on compact subsets
of G, it follows that the family flog un0(z)g is compact in D0(G). Therefore there
exists a subsequence log un00(z) which converges in D0(G) (and also in L1
loc
(G)) to
some subharmonic function v(z) in G.
Note that for any compact set K � G and for any " > 0 we have the following
inequality:Z
K
jmaxflog un0(z); log "g �maxfv(z); log "gjdxdy �
Z
K
j log un0(z) � v(z)jdxdy:
Hence, the functions maxflog un0(z); log "g converge to the function
maxfv(z); log "g in L1
loc
(G) for any " > 0.
On the other hand, as was shown above, l"(un)(z) converge to l"(u0)(z) in
L1
loc
(G). Thus almost everywhere (and, since the functions are subharmonic,
everywhere)
maxfv(z); log "g = maxflog u0(z); log "g: (9)
Since a set on which a subharmonic function equals to �1 has
Lebesgue measure zero, then "! 0 implies that mes(fz 2 G : v(z) < log "g)! 0,
mes(fz 2 G : log u0(z) < log "g)! 0, and v(z) = log u0(z) almost everywhere, and
hence everywhere. Thus the sequence of the functions log un0(z) converges to the
function log u0(z) in D
0(G) and in L1
loc
(G).
If for some subsequence of the functions log unj (z), "0 > 0 and compact set
K0 2 G Z
K0
j log unj (z)� log u0(z)jdxdy � "0; (10)
then, using the above construction of the sequence unj (z), we have that some
subsequence of the sequence flog unjg converges to log u0(z) in L1
loc
(G), which
contradicts (10). The lemma is proved.
P r o o f o f T h e o r e m 2. From Proposition 3 in [2] it follows that the
inclusion log u 2 WAP (S) implies that inclusion u 2 WAP (S). We are going to
show the opposite inclusion. Let u(z) 2WAP (S) and fhng � R be an arbitrary
sequence. Passing to a subsequence if necessary, we can assume that for some
subharmonic function u0, uniformly in t 2 R,
lim
n!1
Z
S
(u(z + hn)� u0(z))'(z � t)dxdy = 0: (11)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 217
A.V. Rakhnin and S.Yu. Favorov
To prove the theorem it is su�cient to verify that uniformly in t 2 R
lim
n!1
Z
S
log u(z + hn + t)'(z)dxdy =
Z
S
log u0(z + t)'(z)dxdy: (12)
Assuming that this fails, for some " > 0 and some sequence tn !1,
������
Z
S
log u(z + hn + tn)'(z)dxdy �
Z
S
log u0(z + tn)'(z)dxdy
������ � ": (13)
Here u0(z) is a logarithmic subharmonic function with u0(z) 2WAP (S). Passing
to a subsequence and using almost periodicity of the function u0(z), we can assume
that
lim
n!1
Z
S
u0(z + tn)'(z)dxdy =
Z
S
v(z)'(z)dxdy (14)
for some subharmonic in the strip S function v(z). Since the limit in (11) is
uniform in t 2 R, (14) implies
lim
n!1
Z
S
u(z + hn + tn)'(z)dxdy =
Z
S
v(z)'(z)dxdy:
Now Lemma 4 implies that both integrals in (13) have the same limitR
log v(z)'(z)dxdy, when n ! 1, which is impossible. Thus (12) holds and
Theorem 2 is proved.
P r o o f o f T h e o r e m 3. Without loss of generality, we can assume that
S is a strip with �nite width. Let S0 be an arbitrary substrip, S0 �� S. Since
the function u(z) is almost periodic, its Riesz measure � := 1
2�
�u is also almost
periodic in the sense of distributions.
Denote
K(w) =
1
2
log je�
w
2
� 1j;
where
0 <
<
�
max
y1;y22ImS
(y1 � y2)2
:
Note that the kernel K(w) is a subharmonic function which is bounded from
above in S and its restriction to S0 satis�es the equation
�K(w) = 2�Æ(w); (15)
218 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
Subharmonic almost periodic functions
where Æ(w) is a standard Dirac measure. Denote
V (z) =
Z
S
K(w � z)'(Imw)d�(w); (16)
where ' � 0 is a test function on ImS such that '(y) = 1 for y 2 ImS0.
Denote Pn = f(n � 1=2; n + 3=4) � ImSg � S. We are going to show that
V (z) is a subharmonic function in every Pn. Fixing n0 2 Z, we have
Z
S
K(w � z)'(Imw)d�(w) =
Z
[n0�1;n0+1)�ImS
K(w � z)'(Imw)d�(w)
+
X
n2Znfn0�1;n0g
Z
[n;n+1)�ImS
K(w � z)'(Imw)d�(w): (17)
Every term in the right hand side of (17) is obviously a subharmonic function.
For Rew 2 [n; n+ 1), Imw 2 supp', z 2 Pn0 , n 6= n0, n 6= n0 � 1 we have
���e�
(w�z)2
��� = e�
(Rew�Rez)2+
(Imw�Imz)2 � e��
(jn�n0j�3=4)2 :
Thus
X
n2Znfn0�1;n0g
�������
Z
[n;n+1)�ImS
K(w � z)'(Imw)d�(w)
�������
�
X
n2Znfn0�1;n0g
sup
z2Pn0
sup
w2[n;n+1)�supp'
����12 log j1� e�
(z�w)
2
j
�����([n; n+1)� supp'):
Since the measure � is almost periodic, �([n; n + 1) � supp') is bounded from
above uniformly in n (see [2]), and therefore the series (17) converges uniformly
in z 2 Pn0 and the function V (z) is subharmonic in Pn0 , and also in S.
Now we are going to show that the function V (z) is subharmonic almost
periodic in S. We consider a test function (z) on S and verify that the function
f(t) =
Z
S
V (z) (z � t)dxdy
is uniformly almost periodic on the real axis. We have
f(t) =
Z
S
0
@Z
S
K(w � z) (z)dxdy
1
A'(Imw)d�(w + t):
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 219
A.V. Rakhnin and S.Yu. Favorov
Note that the function
(w) :=
Z
S
K(w � z) (z)dxdy
is continuous in S, because the di�erence K(w) � log jwj is continuous in some
neighborhood of zero. Moreover, (w) = O(e�
jwj
2
) when jRewj ! 1.
Since the values �([n; n+ 1]� ImS0) are uniformly bounded in n, thenZ
S
'(Imw + Imz)d�(w + z)
1 + jwj2
� C1 <1 (18)
uniformly in z 2 S0.
We �x " > 0 and choose a test function �(t), 0 � �(t) � 1 on R, and such
that �(Rew) = 1 on the set�
w : j (w)j >
"
C1(1 + jwj2)
�
:
For all t 2 R we have
f(t) =
Z
S
(w)�(Rew)'(Imw)d�(w + t)
+
Z
S
(w)(1 � �(Rew))'(Imw)d�(w + t):
Property (18) implies that the second integral in the equality is not greater than "
for all t 2 R. Since � is an almost periodic measure, the �rst integral is an almost
periodic function, and if � is an "-almost period, then it is a 2"-almost period
for f . Thus, the function V (z) is a subharmonic almost periodic, and in addition
(15) implies that �V (z) = 2�'(y)�(z) in the sense of distributions. Consider the
function
H(z) := V (z)� u(z):
This function is harmonic and almost periodic in the sense of distributions in S0.
Let ' � 0 be a test function in the disk B("; 0), which depends only on jzj and
such that
R
'(z)dxdy = 1. Since the convolution
R
H(z)'(z + �)dxdy is equal to
H(�) in some strip S1 �� S0, then the remark after De�nition 2 implies uniform
almost periodicity of the function H(z) in S1. So its Fourier�Bohr coe�cients are
continuous in ImS1 and, since " is arbitrary, in ImS0. Thus it is enough show
that the Fourier�Bohr coe�cients of V (z),
a�(V; y) =M(V e�i�x; y) = lim
T!1
1
2T
TZ
�T
V (x+ iy)e�i�xdx;
220 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
Subharmonic almost periodic functions
are continuous.
We �x " > 0. We have
K(w) = maxfK(w);�2 logNg+minfK(w) + 2 logN; 0g = K1(w) +K2(w);
where N <1 will be chosen later. Denote
V1(z) :=
Z
S
K1(z � w)'(Imw)d�(w);
V2(z) :=
Z
S
K2(z � w)'(Imw)d�(w):
Since for j
w2j < 1=2 we have
K(w) = 1=2 log j1� 1 +
w2 �
2w4
2!
+ : : : j = log jwj + �(w);
where �(w) is a continuous function, then K2(w) = 0 for jwj � Æ0 > 0 and N
su�ciently large. Moreover, if j�(w)j � logN , then for all w 2 C and y 2 ImS,
TZ
�T
K2(z � w)dx =
TZ
�T
minflog jx� u+ i(y � v)j+ �(z � w) + 2 logN; 0gdx
�
1Z
�1
minflog jNx�Nuj; 0gdx = �
C
N
; (19)
with some constant C, 0 < C < 1. Now using the property that
�([n; n+ 1]� supp') are bounded and the fact thatK2(z�w) = 0 for jz�wj � Æ0,
we have that for all T
������
1
2T
TZ
�T
V2(z)e
�ix�dx
������ �
Z
jRewj�T+Æ0
1
2T
TZ
�T
jK2(z � w)jdxd�(w)
�
C
2TN
Z
jRewj�T+Æ0
'(Imw)d�(w) �
C2
N
� "; (20)
if N is su�ciently large.
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 221
A.V. Rakhnin and S.Yu. Favorov
Further, since K1(w) = O(e�
jwj
2
) for jRewj ! 1, then one can choose a test
function 0 � �(t) � 1 on R such that �(Rew) = 1 on the set
�
w : jK1(w)j >
"
C1(1 + jwj2)
�
;
where C1 is the constant from (18). We have
V1(z) =
Z
S
K1(w)�(Rew)'(Imw + Imz)d�(w + z)
+
Z
S
K1(w)(1 � �(Rew))'(Imw + Imz)d�(w + z) = V3(z) + V4(z):
From the choice of the function � it follows that
jV4(z)j � "; for z 2 S0: (21)
Since the kernel K1(w) is continuous and the family of shifts of a test function in
ImS0 is a compact set, then (see the remark to De�nition 2) the function V3(z)
is uniformly almost periodic in S0 and it has continuous in ImS0 Fourier�Bohr
coe�cients (see [1, p. 145]). Thus, if y1; y2 2 ImS0 and jy1 � y2j � Æ("), then (20)
and (21) imply
ja�(V; y1)� a�(V; y2)j � ja�(V3; y1)� a�(V3; y2)j+ ja�(V4; y1)j+ ja�(V4; y2)j
+ja�(V2; y1)j+ ja�(V2; y2)j � 5":
Thus a�(V; y) are continuous. The theorem is proved.
P r o o f o f T h e o r e m 4. For Pm(z) we choose Bohner�Fejer sums of
the function u(z)
Pm(z) := lim
T!1
1
2T
TZ
�T
u(z + t)�(m)(t)dt =
X
k
(m)
�
a�(u; Imz)e
i�Rez:
Here �(m)(t) is a sequence of Bohner�Fejer kernels (see. [3, p. 69]), and the set
fk
(m)
�
: k
(m)
�
6= 0g is �nite for every m. Note that, according to Theorem 3, the
functions a(u; y) are continuous in y 2 ImS.
We are going to show that Pm(z) are subharmonic.
Note that the kernels �(m)(t) are nonnegative, bounded, and
lim
T!1
1
2T
TZ
�T
�(m)(t)dt = 1:
222 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
Subharmonic almost periodic functions
Also note that the subharmonic almost periodic function u(z) is bounded from
above in any subset S0 �� S. Thus, using Fatou's lemma, for any m = 1; 2; : : : ,
z 2 S, and su�ciently small �,
1
2��
2�Z
0
Pm(z + �ei')d' � lim
T!1
1
2T
TZ
�T
1
2��
2�Z
0
u(z + �ei' + t)�(m)(t)d'dt
� lim
T!1
1
2T
TZ
�T
u(z + t)�(m)(t)dt = Pm(z):
As it is shown in [6], for any test function '(z) in S and for some (depending
only on the spectrum of the function u(z)) sequence of Bohner�Fejer sums, for
m!1, uniformly in t 2 R
Z
S
Pm(z)'(z + t)dxdy !
Z
S
u(z)'(z + t)dxdy: (22)
Now we are going to verify that it implies the convergence in the topology
de�ned by seminorms d[�;�], �; � 2 ImS. Indeed, if it is not true, then for some
sequence xm !1 and some �; � 2 ImS, "0 > 0,
sup
y2[�;�]
1Z
0
ju(xm + iy + t)� Pm(xm + iy + t)jdt � "0:
Since u 2 StAP (S), then, passing to a subsequence if necessary, one can assume
that functions u(z+xm) converge to some function v 2 StAP (S) with respect to
metric d[�;�], and therefore
sup
y2[�;�]
1Z
0
jPm(xm + iy + t)� v(t+ iy)jdt � "0=2: (23)
Moreover, according to Theorem 2,Z
S
u(z + xm)'(z)dxdy !
Z
S
v(z)'(z)dxdy:
Therefore, by setting t = �xm in (22), we have for any test function '(z)
lim
m!1
Z
S
Pm(z + xm)'(z)dxdy =
Z
S
v(z)'(z)dxdy:
According to Lemma 2, this contradicts to (23). The theorem is proved.
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 223
A.V. Rakhnin and S.Yu. Favorov
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[3] B.M. Levitan, Almost periodic functions. Gostehizdat, Moscow, 1953. (Russian)
[4] A. Rakhnin, On one property of the subharmonic almost periodic functions. �
Visnyk Harkivs'kogo universytetu 53 (2003), No. 602, 24�30. (Russian)
[5] L.I. Ronkin, Jessen's theorem for holomorphic almost periodic functions in tube
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224 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
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