On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions
We obtain the results analogous of those of [5] on the polynomial asymptotics with arbitrary 0 < ρn < ... < ρ1 < ρ, defning multipolynomial terms.
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Azarin, V. 2016-09-28T17:52:12Z 2016-09-28T17:52:12Z 2007 On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions / V. Azarin // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 5-12. — Бібліогр.: 12 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106434 We obtain the results analogous of those of [5] on the polynomial asymptotics with arbitrary 0 < ρn < ... < ρ1 < ρ, defning multipolynomial terms. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions Article published earlier |
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On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions |
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On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions Azarin, V. |
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On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions |
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On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions |
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On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions |
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On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions |
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on the polynomial asymptotics of subharmonic functions of finite order and their mass distributions |
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Azarin, V. |
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Azarin, V. |
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2007 |
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Журнал математической физики, анализа, геометрии |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We obtain the results analogous of those of [5] on the polynomial asymptotics with arbitrary 0 < ρn < ... < ρ1 < ρ, defning multipolynomial terms.
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1812-9471 |
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https://nasplib.isofts.kiev.ua/handle/123456789/106434 |
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On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions / V. Azarin // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 5-12. — Бібліогр.: 12 назв. — англ. |
| work_keys_str_mv |
AT azarinv onthepolynomialasymptoticsofsubharmonicfunctionsoffiniteorderandtheirmassdistributions |
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2025-11-27T03:05:02Z |
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2025-11-27T03:05:02Z |
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1850795935056003072 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 1, pp. 5�12
On the Polynomial Asymptotics of Subharmonic
Functions of Finite Order and their Mass Distributions
Vladimir Azarin
Department of Mathematics and Statistics, Bar-Ilan University
Ramat-Gan, 52900, Israel
E-mail:azarin@macs.biu.ac.il
Received September 1, 2006
We obtain the results analogous of those of [5] on the polynomial asymp-
totics with arbitrary 0 < �n < : : : < �1 < �, de�ning multipolynomial
terms.
Key words: subharmonic function, asymptotic representation, limit set
for entire and subharmonic functions, topology of distribution.
Mathematics Subject Classi�cation 2000: 30D20, 30D35, 31A05, 31A10.
1. Introduction and the Main Results
1.1. In papers [1�8], it is considered, in particular, the polynomial asymptotics
of subharmonic functions of �nite order � and their mass distributions in terms
of the growth of reminder terms and topology of exceptional sets. Besides, the
exponents �1; : : : ; �n of terms had to satisfy the conditions [�] < �n < : : : < � for
a noninteger �. We are going to represent another point of view by studying the
polynomial asymptotics in D0-topology and a little bit stronger topology and relax
restriction on the exponent to the natural � > �1 > : : : > �n > 0. It occurs that
this change of topology together with the consideration of more narrow class than
in [5] allows to obtain a multiterm asymptotic analog of Levin�P�uger's theory of
completely regular growth and make simpler (in our opinion) formulating of the
results and proofs.
By �D0-topology� we call the topology of the space D0(C n0) of distributions
(i.e., linear bounded functionals) over the basic space D(C n0) of �nite in�nitely
di�erentiable functions. Recall that a sequence uj ! 0, j ! 1 in this space if
the linear functionals
< uj; g >! 0 (1:1:1)
for all g 2 D:
c
Vladimir Azarin, 2007
Vladimir Azarin
About connection between D0-topology and the topology of exceptional sets
for subharmonic functions see [9], [10, Ch. 3].
We also use C1
q;p
*-topology, i.e., the topology of linear functionals over the
basic space C1
q;p
with the convergence de�ned like in (1.1.1). The space C1
q;p
is
one of the in�nitely di�erentiable functions in C n 0 that tends to 1 not faster
then O(jzj�q) as z ! 0 and tends to zero not slower then O(jzj�p) as z !1. Let
us note that this topology is stronger that D0-topology because C1
q;p � D(C n0):
Let u(z) be a subharmonic function in C of normal type with respect to a �nite
order �, i.e.,
0 < �[u] := lim sup
r!1
M(r; u)r�� <1;
where M(r; u) := maxjzj=r u(z).We write u 2 SH(�):
Let � be a mass distribution in C with no mass in zero. It has normal type
with respect to the exponent � if
0 < �[�] := lim sup
r!1
�(Kr)r
��
<1;
where Kr := fz : jzj < rg:We write � 2M(�): De�ne by �u the mass distribution
associated with u: Recall
Borel's Theorem. Let [�] < �: If u 2 SH(�), then � 2M(�) and vice versa.
Let � = [�] := p. Set
ÆR(z; �; p) :=
1
p
Z
j�j<R
<
�
z
�
��
�(d�d�):
This is a family of the homogeneous harmonic polynomial of degree p: Recall in
an equivalent formulation
Lindel�of's Theorem. If u 2 SH(�) then �u 2M(�) and the family fÆRg is
precompact as R!1; and vice versa.
Denote ut(z) := u(tz)t��: The function u(z) 2 SH(�) is called a function of
the completely regular growth (CRG-function) if ut ! h�(z) in D0-topology, as
t!1: Here
h�(z) := r
�
h(ei�) (1:1:2)
and the function h(ei�) is a �-trigonometrically convex function (�-t.c. function)
(see, e.g., [10, Ch. 1, �� 15, 16] ), i.e., it is a 2�-periodic generalized solution of
the equation
h
00 + �
2
h = �(d�); (1:1:3)
where � is a 2�-periodic positive measure.
6 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1
On the Polynomial Asymptotics of Subharmonic Functions of Finite Order
Recall also that �-t.c. function as a distribution is equivalent to a continuous
function and can be represented for noninteger � in the form
h(�) =
1
2� sin��
2�Z
0
� cos �(�� � �) �(d ); (1:1:4)
where the function � cos �(�) is a 2�-periodic extension of the function cos �� from
the interval (��; �) on (�1;1). If �(> 0) is integer, then � must satisfy the
condition
2�Z
0
e
i���(d�) = 0; (1:1:5)
and the representation has the form
h(�) = <fCei�)g+
1
2�
2�Z
0
�(�� ) sin�(�� )�(d ); (1:1:6)
where C is a complex constant, the function � means the 2�-periodic continua-
tion of the function f( ) := from the interval [0; 2�) on (�1;1):
Recall (see [9], [10, Ch.3, � 1]) that �t (do not confuse with �u) is the mass
distribution de�ned by the equality
< �t; g >:= t
��
Z
g(z=t)�(dxdy)
for all g 2 D: It can also be de�ned by the equality
�t(E) := �(tE)t��;
where E is every Borel set and tE is the homothety of E:
Let � > [�]: Recall that the mass distribution � is called regular if
�t ! �(d�)
�r
��1
dr (1:1:7)
in D0-topology as t!1:�(d�) is a measure on the unit circle which is necessarily
positive.
Let � be an integer number p = [�]: Then the mass distribution is called
regular if, in addition to (1.1.7), ÆR(z; �t; p) converges in D
0-topology as t ! 1
for some R:
Since ÆR(z; �t; p) is a homogeneous harmonic polynomial, the convergence in
D0-topology is equivalent to uniform convergence in every bounded domain.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1 7
Vladimir Azarin
In such terms Levin�P�uger's theorem (see [9, Chs. 2, 3], [10, Ch. 3, Th. 3])
may be formulated as follows.
Levin�P�uger's Theorem. If u is a CRG-function, then its mass distribu-
tion is regular and vice versa.
1.2. Let �̂ = f� > �1 > : : : > �n > 0g be a �nite monotonic system of
numbers. We call a function u 2 SH(�) completely �̂-regular if
ut = h� + t
�1��h�1 + : : :+ t
�n��h�n + t
�n��o(1); (1:2:1)
where h� is a �-t.c. function and h�j (z); j = 1; 2; : : : ; n, are of the form of (1.1.2)
with the corresponding h's being the di�erences of �j-t.c. functions. Therefore h�j
can be represented in the form of (1.1.4) or (1.1.6) with �'s being the functions
of bounded variation. Besides, o(1) ! 0 in D0 topology.
Let � > [�] and �j 2 ([�]; �), j = 1; 2; : : : ; n. We call � 2M(�) �̂-regular if
�t = �(�) +
j=nX
j=1
t
�j���(�j) + t
�n��o(1)) (1:2:2)
as t!1, where
�(�) = ��(d )
�r
��1
dr; (1:2:3)
with �� positive and summable, and �(�j), j = 0; 1; : : : ; n, are of the same form
as � = �j , j = 0; 1; : : : ; n, and arbitrary �(�j)'s that are the functions of bounded
variation on the circle.
If o(1) ! 0 in D0-topology, then � is �̂-regular in D0-topology. However it is
possible to say that � is �̂-regular in other topology if o(1) ! 0 in this topology.
Theorem 1.2.1. Let � > [�] and [�n; �] \ N = ;: If u is completely �̂-regular
in D0-topology then its mass distribution � is �̂-regular in D0 topology. If � is
�̂-regular in C1
p;p+1*-topology, then u is completely �̂-regular in D0-topology.
Let us notice that the classical Levin�P�uger theorem of completely regular
growth function for noninteger � can be obtained from here by using the following
Proposition 1.2.2. Let � 2 M(�) and �t ! �(�) in D0 as t ! 1: Then
the same holds in C1
p;p+1�.
We suppose further that � is an exponent of the convergence of �.
Let us consider the situation, when �̂ consists of noninteger numbers, but
the interval (0; �) contains integer numbers.
Theorem 1.2.3. Let ut have the representation
ut = h� + t
�1��h�1 + : : :+ t
�n��h�n +
[�]X
1
<fakz
kgtk�� + t
�n��o(1); (1:2:4)
8 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1
On the Polynomial Asymptotics of Subharmonic Functions of Finite Order
where o(1)! 0 in D0
:
Then
�t = �(�) +
j=nX
j=1
t
�j���(�j) + t
�n��o(1) (1:2:5)
with o(1)! 0 in D0
:
The inverse theorem is the following
Theorem 1.2.4. Let u 2 SH(�) and its mass distribution have the represen-
tation (1.2.5) with o(1) ! 0 in C1
p;p+1* and
2�Z
0
e
ik���j
(d�) = 0 (1:2:6)
for all k, � > k > �j.
Then (1.2.4) holds for ut with o(1) ! 0 in D0
:
Let us notice that the conditions (1.2.6) are not necessary for the validness of
(1.2.4).
The similar theorems can be formulated for the case when � or some of �j are
integers.
I am grateful to Prof. V. Logvinenko for his valuable notes.
2. Proofs
2.1. Consider the case when � > [�] and [�n; �] \ N = ;. Let ut have the re-
presentation (1.2.1) and the remainder term be o(1) in D0-topology. Applying to
(1.2.1), the operator (1=2�)� (here � is the Laplace operator) we obtain (1.2.2),
as (1=2�)�ut = �t; (1=2�)�h�j = ��j
(d�), j = 0; : : : ; n, and (1=2�)�o(1) = o(1)
since the Laplace operator is continuous in D0-topology. The �rst assertion of
Th. 1.2.1 is proved.
Let (1.2.2) hold with o(1) in C1
p;p+1*. Apply to it the operator Ad�
�
which is
conjugated to
Ad�[�] :=
Z
Cn0
H(z=�; [�]) � (dxdy)
that acts from D to C1p; p+ 1: By de�nition, for g 2 D we have
< Ad
�
��t; g >=< �t; Ad�[g] > :
Now substitute (1.2.2) for �t. The integral of the �rst n terms of (1.2.2 ) are, in
fact, the �rst n terms of (1.2.1). Let us verify it.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1 9
Vladimir Azarin
We have
< �(�j); Ad�[g] >z=
Z
g(z)dxdy
Z
H(z=rei ; p)�j(d )�jr
�j�1
dr:
Counting the inner integral on dr (see, [11, Ch. 1, � 17, footnote 21]), we obtain
1Z
0
H(z=rei ; p)�jr
�j�1
dr =
1
2�j sin��j
� cos �(arg z � � �)jzj�j : (2:1:1)
Hence, using (1.1.4), we obtain
< ��j ; Ad�[g] >z=< h�j :g > : (2:1:2)
The last term is t�n��o(1) where o(1) is understood in C1
p;p+1*. The function
Ad�[g] is a canonical potential of the function g 2 D: Thus Ad�[g] 2 C
1
p;p+1.
Therefore < o(1); Ad�[g] >z! 0 as t ! 1: This proves the second assertion of
Th. 1.2.1.
2.2. Let us prove Proposition 1.2.2.
P r o o f. Let g 2 C
1
p;p+1: Let �1, �2, �3 be a partition of unity by in-
�nitely di�erentiable functions, such that supp �1 � (0; �), supp �2 � (�=2; 2R),
supp �3 � (R;1): Then
Z
C
g(z)�t(dxdy) = I1(t) + I2(t) + I3(t);
where
Ij(t) =
Z
C
g(z)�j(jzj)�t(dxdy); j = 1; 2; 3:
The �rst integral has the estimate
jI1(t)j � lim
Æ!0
�Z
Æ
Cr
�p
�t(dr);
because g is O(jzj�p) as z ! 0: Integrating by parts, we obtain
I1(t) � C
2
4�t(�)���p + lim
Æ!0
�Z
Æ
r
�p�1
�t(r)(dr)
3
5 :
10 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1
On the Polynomial Asymptotics of Subharmonic Functions of Finite Order
Since �(r) � Cr
�, also �t(r) � Cr
�
: Thus
I1(t) � C�
��p (2:2:1)
uniformly with respect to t:
In the same way we obtain
I3(t) � CR
��p�1 (2:2:2)
uniformly with respect to t.
Since �t ! �� in D
0 and g�2 2 D, we have
I2(t)!
Z
C
g(z)�2(jzj)��(dxdy); t!1: (2:2:3)
Moreover, (2.2.1),(2.2.2), and (2.2.3) imply that
< g; �t >!< g; �(�) >
for every g 2 C1
p;p+1 because � can be chosen to be arbitrarily small and R can be
selected to be arbitrarily large.
For proving Th. 1.2.3 we should only repeat the �rst part of the proof of
Th. 1.2.1.
2.3.
P r o o f o f T h e o r e m 1.2.4. As in the proof of Th. 1.2.1 we apply
the operator Ad�� to �t and evaluate < ��j ; Ad�[g] >z. Because of (1.2.3),
< �(�j); Ad�[g] >z=< �jr
�j�1
; < ��j
; Ad�[g] >�>r;
where
< ��j
; Ad�[g] >�:=
2�Z
0
Ad�[g](re
i�)��j
(d�):
Changing the order of integration and using (1.2.6) and (2.1.1), we obtain
< �(�j); Ad�[g] >z=< �(�j); Ad�j [g] >z=< h�j ; g > :
As it was explained in the proof of Th. 1.2.1, < o(1); Ad�[g] >! 0: Thus
Ad
�
��t = h� + t
�1��h�1 + :::+ t
�n��h�n + o(1)t�n��: (2:3:1)
By Adamar's theorem (see, e.g., [12, Ch. 4.2])
u(z)�Ad
�
�
�(z) =
[�]X
k=0
<fakz
kg: (2:3:2)
Thus (2.3.1) and (2.3.2) imply (1.2.4).
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1 11
Vladimir Azarin
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12 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1
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