Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane
In the paper, a multi-term asymptotic representation for distribution function of the Riesz measure of subharmonic function in the plane is considered. It is shown that the "smallness" of the reminder term of asymptotic representation does not guarantee the bounded variation with respect t...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2009
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irk-123456789-1065292016-10-01T03:01:40Z Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane Agranovich, P. In the paper, a multi-term asymptotic representation for distribution function of the Riesz measure of subharmonic function in the plane is considered. It is shown that the "smallness" of the reminder term of asymptotic representation does not guarantee the bounded variation with respect to the angle variable of all terms of this asymptotics, and the conditions for this property to be held are given. 2009 Article Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane / P. Agranovich // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 3-11. — Бібліогр.: 6 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106529 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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In the paper, a multi-term asymptotic representation for distribution function of the Riesz measure of subharmonic function in the plane is considered. It is shown that the "smallness" of the reminder term of asymptotic representation does not guarantee the bounded variation with respect to the angle variable of all terms of this asymptotics, and the conditions for this property to be held are given. |
| format |
Article |
| author |
Agranovich, P. |
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Agranovich, P. Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane Журнал математической физики, анализа, геометрии |
| author_facet |
Agranovich, P. |
| author_sort |
Agranovich, P. |
| title |
Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane |
| title_short |
Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane |
| title_full |
Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane |
| title_fullStr |
Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane |
| title_full_unstemmed |
Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane |
| title_sort |
multi-term asymptotic representations of the riesz measure of subharmonic functions in the plane |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106529 |
| citation_txt |
Multi-Term Asymptotic Representations of the Riesz Measure of Subharmonic Functions in the Plane / P. Agranovich // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 3-11. — Бібліогр.: 6 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT agranovichp multitermasymptoticrepresentationsoftherieszmeasureofsubharmonicfunctionsintheplane |
| first_indexed |
2025-07-07T18:36:26Z |
| last_indexed |
2025-07-07T18:36:26Z |
| _version_ |
1837014331833712640 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 1, pp. 3�11
Multi-Term Asymptotic Representations of the Riesz
Measure of Subharmonic Functions in the Plane
P. Agranovich
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:agranovich@ilt.kharkov.ua
Received August 2, 2007
In the paper, a multi-term asymptotic representation for distribution
function of the Riesz measure of subharmonic function in the plane is con-
sidered. It is shown that the "smallness" of the reminder term of asymptotic
representation does not guarantee the bounded variation with respect to the
angle variable of all terms of this asymptotics, and the conditions for this
property to be held are given.
Key words: Subharmonic function, the Riesz measure, multi-term asymp-
totic representation, function of bounded variation.
Mathematics Subject Classi�cation 2000: 31A05, 31A10, 26A45.
One of the most important problems in the function theory is a question on
the connection between the regularity in distribution of zeros (masses) of an entire
(subharmonic) function and its behavior at in�nity. A number of problems in the
�elds close to complex analysis, contiguous areas of mathematics, physics and
radiophysics lead to this question.
In the 30s of the previous century B. Levin (Ukraine) and A. P��uger (Switzer-
land) simultaneously and independently constructed the function theory of com-
pletely regular growth. The theory describes the connection between the distribu-
tion functions of zeros and the entire function in the terms of one-term asymptotic
representations?.
But sometimes either the behavior of function or the growth of distribution
function is given by multi-term asymptotic representation.
Let us recall these notions.
?In [6] there is an extensive bibliography.
c
P. Agranovich, 2009
P. Agranovich
De�nition 1.? Let � be a measure in the plane. Its distribution function
�(t; �) is equal to measure � of sector f(r; �) : 0 < r � t; 0 � � < �g:
De�nition 2. A multi-term (polynomial) asymptotic representation of func-
tion f(t; �); t > 0; � 2 [0; 2�); as t!1; is
f(t; �) = �1(�)t
�1 +�2(�)t
�2 + : : :+�n(�)t
�n + '(t; �);
where �j; j = 1; 2; : : : ; n; are real functions; 0 � [�1] < �n < �n�1 < : : : < �1,
and function '(t; �) is small in a certain sense compared to the previous term.
Let �(t; �) be a distribution function of positive measure � in the plane.
We suppose that �(t; �) has a multi-term asymptotics, i.e.,
�(t; �) = �1(�)t
�1 +�2(�)t
�2 + : : :+�n(�)t
�n + '(t; �); t > 0; � 2 [0; 2�);
where �1(�) > 0; �j; j = 2; 3; : : : ; n; are real functions; 0 � [�1] < �n < �n�1 <
: : : < �1, and function '(t; �) is small in a certain sense compared to the previous
term.
It is known that in the case of polynomial asymptotics (n > 1) the properties
of the �rst term di�er essentially from other terms of this asymptotics. By [1]
and [2] the �rst term of asymptotics is a monotone nondecreasing function of �
for any �xed t. At the same time the second and the next terms of asymptotics
may have unbounded variation. Thus it is natural to study the in�uence of the
reminder term on the properties of the main terms of asymptotics. This problem
is the central item of the paper.
The example below is taken from [2] wherein there is some inaccuracy.
E x a m p l e 1. Let 0 � [�1] < �2 < �1 < [�1] + 1;
!j =
jX
k=1
k
�1�(�1��2);
c!
1
= 2�;
cj = c!j, j = 1; 2; : : : ; c0 = 0; c0
j
= cj�1 +
c
2
j
�1�(�1��2), j = 1; 2; : : : : Notice that
c
0
j
is the middle of the interval (cj�1; cj):
For � 2 [0; 2�] de�ne a continuous function �2 as follows:
�2(cj) = �2(2�) = 0; j = 0; 1; : : : ;
�2(c
0
j) =
1
j
; j = 1; 2; : : : :
Let �2 be a linear function on the other parts of interval [0; 2�].
?For the case of discrete measures the analogous notion is in [4].
4 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Multi-Term Asymptotic Representations of the Riesz...
Evidently, the total variation V 2�
0 f�2g of �2 is 1.
For each �xed t � 0 let �(t; �) be a characteristic function of the segment
[c![t]; 2�].
We put
'(t; �) :=
�
0; 0 < t � 1; 0 � � < 2�;
��2(�)t
�2�(t; �); t > 1; 0 � � < 2�:
(1)
Now we divide the set
C n f(t; �) : t � 1; 0 � � < 2�g
into "curvilinear" rectangulars in the following way. First, we represent the set
as a union of annuli
1[
j=1
f(t; �) : j < t � j + 1; 0 � � < 2�g:
Then we cut the j�ring into "curvilinear" rectangulars:
B
�
(j; l) = fj � t < j + 1; cl � � < c
0
l+1g
and
B
�(j; l) = fj � t < j + 1; c0l+1 � � < cl+1g;
l = 0; 1; : : : :
Consider three measures in the plane de�ned by the following densities with
respect to measure dtd�, respectively:
p1(t; �) =
�
0; 0 < t � 1; 0 � � < 2�;
�1ht
�1�1; t > 1; 0 � � < 2�;
where the positive constant h will be chosen later;
p2(t; �) =
8<
:
0; 0 < t � 1; 0 � � < 2�;
�2
2
c
l
�1��2t
�2�1; t > 1; cl�1 < � � c
0
l
;
��2
2
c
l
�1��2t
�2�1; t > 1; c0
l
< � � cl;
l = 1; 2; : : : :
p3(t; �) =
8>><
>>:
0; 0 < t � 1; 0 � � < 2�;
0; j < t � j + 1; 0 � � � cj�1;
��2
2
c
l
�1��2t
�2�1; (t; �) 2 B
�
(j; l);
�2
2
c
l
�1��2t
�2�1; (t; �) 2 B�(j; l);
j = 1; 2; : : : ; l = j � 1; j; : : : :
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 5
P. Agranovich
Consider the function
p = p1 + p2 + p3:
Notice that on the set
1S
k=j�1
(B
�
(j; k)
S
B
�(j; k)) the function p is equal to p1 and
on the set
j�2S
k=1
(B
�
(j; k)
S
B
�(j; k)):
p = p1 + p2:
It is not di�cult to show that p is a nonnegative function if h > 4
c
:
Let � be a positive measure corresponding to density p with respect to measure
dtd�: It is easy to see that the distribution function �(t; �) of this measure has
the form
�(t; �) = h�t
�1 +�2(�)t
�2 + '(t; �); (2)
where ' is de�ned by ( 1 ).
We have the following estimate for ' :
'(t; �) = O(t�1�1); t!1;
uniformly for � 2 [0; 2�]:
So, we have constructed the distribution function of the Riesz measure of
subharmonic function in the plane with the two-term asymptotic representation.
The second main term of this asymptotics �2 has the in�nite variation on [0; 2�]:
R e m a r k 1. It is easy to see that essential circumstance in the construction
of this example is the following fact. The slope of �2 is not less than �2
c
j
�1��2
on the interval (cj�1; cj):
Let us modify this example a little. Put
�2(c
0
n) =
n; n = 1; 2; : : : ;
where 0 <
n <
1
n
and
1X
n=1
n =1:
If we repeat the construction of Ex. 1 with these data, then we again obtain
a distribution function of the Riesz measure of subharmonic function in the plane.
This distribution function has a two-term asymptotic representation with the
analogous conclusions for function �2. The reminder term of this asymptotics
satis�es the estimate
j'(t; �)j = O(j�1
j); j � t < j + 1; j !1;
uniformly for � 2 [0; 2�]:
6 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Multi-Term Asymptotic Representations of the Riesz...
Now we will show that it is possible to reduce essentially the growth of the
reminder term '(t; �) of a multi-term asymptotic representation, nevertheless,
the second main term of this asymptotics will still have the in�nite variation with
respect to the angle variable.
E x a m p l e 2. Consider the convergent series
1X
k=1
1
k(k + 1)
:
Further we will preserve the notations of Example 1.
Divide each set
ij := (cj�1; cj); j = 1; 2; : : : ;
into intervals by points:
cj;m := cj�1 + cj
�(�1��2)
mX
k=j
1
k(k + 1)
; m = j; j = 1; : : : :
On the segment [0; 2�] we de�ne a continuous function �2 in the following
way:
�2(0) = �2(2�) = �2(cj) = �2(cj;m) = 0;
j = 1; 2; : : : ; m = j; j + 1; : : : ;
�2(c
0
j;m) =
1
(m+ 1)(m+ 2)
;
j = 1; 2; : : : ; m = j; j + 1; : : : ;
where c0
j;m
is the middle of the interval (cj;m; cj;m+1):
�2 is taken to be a linear function on the rest of the parts of segment [0; 2�].
The maximum value of �2 is equal to 1=j(j + 1) on segment (cj�1; cj), j =
1; 2; : : : .
Simple calculations show that the variation of �2 is equal to 1=j on segment
(cj�1; cj), j = 1; 2; : : : . So, the function �2 has the in�nite variation on [0; 2�]:
Let us de�ne the functions '(t; �) and �(t; �) by formulas (1) and (2), respec-
tively.
On each interval ij there is a sequence of intervals on which �2(�) is a de-
creasing linear function. Notice? that on these intervals the slope of �2 is equal
to �2
c
j
�1��2 :
Now we carry out the construction to Example 1. It is clear how to choose
three densities of measures in the plane so that their sum is a density of non-
negative measure � with respect to measure dtd� in the plane. It is easy to see
?See Remark 1 .
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 7
P. Agranovich
that �(t; �) is the distribution function of this measure. The reminder term ' of
this asymptotics satis�es the estimate
j'(t; �)j = O(j�1�2); j !1;
uniformly for � 2 [0; 2�]:
Moreover, the analysis of the constructions in Ex. 1 and Ex. 2 shows that it
is possible to reduce the growth of the remainder term and to obtain the same
conclusion about the behavior of the main terms of asymptotics.
We have demonstrated that the "smallness" of the reminder term of asymp-
totic representation does not guarantee the bounded variation with respect to the
angle variable of all terms of this asymptotics.
Moreover, the above examples show that if the distribution function of the
Riesz measure and the �rst main term of asymptotics satisfy the Lipschitz con-
dition? with respect to the angle variable at some point, then this condition does
not necessarily hold for other terms of asymptotics. In fact, it is easy to see that
in our examples this e�ect appears at point � = 2�:
There is a special situation when the boundedness of variation can be claimed
for all terms. This is the case
'(t; �) = '1(t)'2(�):
Theorem 1. Let a distribution function of measure � have the representation
�(t; �) =
nX
j=1
�j(�)t
�j + '(t; �); t > 0; � 2 [0; 2�]; (3)
where �1 is a monotone nondecreasing function, and '(t; �) = '1(t)'2(�) such
that for some q � 1
2TZ
T
j'1(t)j
q
dt = o(T �nq+1); T !1: (4)
Then each of asymptotic representation (3) is a function of bounded variation.
To prove this theorem we will use the following auxiliary statements about
the determinants of a speci�c type.
?Recall that function f(x) satis�es the Lipschitz condition in some point xo if there are such
positive numbers A and Æ that
jf(xo)� f(y)j � Ajxo � yj;
for jxo � yj < Æ:
8 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Multi-Term Asymptotic Representations of the Riesz...
Lemma 1. ([5, vol. 2, V, probl. 76]) Let 0 < �n < �n�1 < : : : < �2 < �1 and
0 < �1 < �2 < : : : < �n: Then the determinant
��������
�
�n
1 �
�n�1
1 : : : �
�1
1
�
�n
2 �
�n�1
2 : : : �
�1
2
: : : : : : : : : : : :
�
�n
n �
�n�1
n : : : �
�1
n
��������
is positive.
Lemma 2. Let 0 < �n�1 < : : : < �2 < �1 and �k > 0; �k ! +1: If a
function
(t) satis�es the estimate
(t) = o(t�n�1); t!1;
then it is possible to choose n numbers �kj
, j = 1; 2; : : : ; n; from the sequence
fakg such that the determinant
A =
��������
(�k1
) �
�n�1
k1
: : : �
�1
k1
(�k2
) �
�n�1
k2
: : : �
�1
k2
: : : : : : : : : : : :
(�kn) �
�n�1
kn
: : : �
�1
kn
��������
6= 0:
P r o o f. Without loss of generality, one may suppose that j
(t)j=t�n�1 tends
to zero monotonically as t!1:
We will use the induction for the proof of this lemma. We may choose two
numbers �k1
and �k2
such that the determinant
����
(�k1
) �
�n�1
k1
(�k2
) �
�n�1
k2
����
does not equal zero. It follows from the conditions for numbers �k, k = 1; 2; and
the function
(t):
Assume this lemma is true for the determinants of order at most n � 1: Let
us use the Laplace expansion of determinant A along the last column. In virtue
of the assumption of induction the last element of this column �
�1
kn
is multiplied
by nonzero minor. Taking into account the inequalities for the orders �j , j =
1; 2; : : : ; n�1; we can conclude that in the sequence f�kg there is such a su�ciently
large number �n that the determinant A 6= 0. The lemma is proved.
Now we return to our theorem.
From (4) we get such a sequence of points fskg
1
k=1 that lim
k!1
sk = +1 and
'1(sk) = o(s�n
k
); k !1:
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 9
P. Agranovich
In virtue of Lemma 2 we can choose n points s1; : : : ; sn from this sequence such
that the determinant ��������
s
�2
1 s
�3
1 : : : s
�n
1 '1(s1)
s
�2
2 s
�3
2 : : : s
�n
2 '1(s2)
: : : : : : : : : : : : : : :
s
�2
n s
�3
n : : : s
�n
n '1(sn)
��������
6= 0:
Substituting these points sk, k = 1; : : : ; n; in (3) we obtain the system of the
linear equations with non-vanishing determinant. Consequently, every term of
asymptotics (3) is the function of bounded variation with respect to variable �.
The theorem is proved.
Now we consider the case of the remainder term of general form.
Theorem 2. Let a distribution function of measure � have representation
(3), where �1 is a monotone nondecreasing function, and there are t1 < t2 <
: : : < tn�1 such that the remainder term '(tj ; �), j = 1; : : : ; n � 1; is a function
of bounded variation.
Then all terms of asymptotic representation (3) are the functions of bounded
variation.
P r o o f. Substituting the values tj, j = 1; : : : ; n� 1; in (3) we obtain the
system of the linear equations
nX
j=2
�j(�)t
�j
k
= �(tk; �)��1(�)t
�1
k
; k = 1; : : : ; n� 1:
In view of Lemma 1 the determinant of this system
��������
t
�2
1 t
�3
1 : : : t
�n
1
t
�2
2 t
�3
2 : : : t
�n
2
: : : : : : : : : : : :
t
�2
n t
�3
n : : : t
�n
n
��������
is not zero. So, it is easy to see that the bounded variation of the functions
�(tj ; �)��1(�)t
�1
j
; j = 1; : : : ; n� 1;
implies the bounded variation of the functions �k; k = 2; : : : ; n: Hence the re-
mainder term '(t; �) is also the function of bounded variation with respect to �
for any t. The theorem is proved.
R e m a r k 2. Notice that the above examples show that any "smallness" of
the remainder term does not retain di�erential properties of the functions �(t; �)
and �1 for other terms of asymptotics, even the Lipschitz condition. At the same
10 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
Multi-Term Asymptotic Representations of the Riesz...
time, the ful�lment of conditions of these theorems guarantees that the functions
�j; j = 2; 3; : : : ; n; and '(t; �) are di�erentiable with respect to � at those points,
where the functions �1 and �(t; �) are di�erentiable.
Thus for the asymptotic representations of measure distribution functions we
have found the su�cient conditions on the remainder term that guarantee the
boundedness of variation and the di�erentiability with respect to the angle vari-
able of all terms of this asymptotics.
Consider now the measure � that satis�es the conditions of Theorems 1 or 2.
It is known [1] that outside of any exceptional set the subharmonic function u(rei�)
corresponding to � has the following asymptotics:
u(rei�) =
nX
j=1
Hj(�)r
�j + (rei�);
where
Hj(�) =
�
sin��j
�Z
��2�
cos �j(� � �� �)d�j(�); j = 1; 2; : : : ; n:
Obviously, from our theorems we obtain that every term of this asymptotics,
starting from the second one, is a Æ-subharmonic function.
This special case has been considered recently in the paper [3].
Acknowledgements. The author is grateful to the reviewer for very useful
remarks.
References
[1] P.Z. Agranovich and V.N. Logvinenko, Multi-Term Asymptotic Representation of
a Subharmonic Function in the Plane. � Sib. Mat. J. 32 (1991), No. 1, 1�16.
(Russian)
[2] P.Z. Agranovich and V.N. Logvinenko, Exceptional Sets for Entire Functions. �
Mat. Stud. 13 (2000), No. 2, 149�156.
[3] V. Azarin, On the Polynomial Asymptotics of Subharmonic Functions of Finite
Order and their Mass Distributions. � J. Math. Phys., Algebra, Geom. 3 (2007),
No. 1, 5�12.
[4] B.Ja. Levin, Distribution of Zeros of Entire Functions. AMS, Providence, RI, 1980.
[5] G.P�olya and G.Szeg�o, Problems and Theorems in Analysis. Springer Verlag, Berlin,
1998.
[6] L.I. Ronkin, Functions of Completely Regular Growth of Several Variables. Kluwer
Acad. Publ., Dordrecht, 1992.
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