skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Fourier series and {delta}-subharmonic functions of finite {gamma}-type in a half-plane

Abstract

Let {gamma}(r) be a growth function and let v(z) be a proper {delta}-subharmonic function in the sense of Grishin in a complex half-plane, that is v=v{sub 1}-v{sub 2}, where v{sub 1} and v{sub 2} are proper subharmonic functions (limsup{sub z{yields}}{sub t}v{sub i}(z){<=}0, for each real t, i=1,2), let {lambda}={lambda}{sub +}-{lambda}{sub -} be the full measure corresponding to v and let T(r,v) be its Nevanlinna characteristic. The class J{delta}({gamma}) of functions of finite {gamma}-type is defined as follows: v element of J{delta}({gamma}) if T(r,v){<=}A{gamma}(Br)/r for some positive constants A and B. The Fourier coefficients of v are defined in the standard way. The central result of the paper is the equivalence of the following properties: (1) v element of J{delta}({gamma}); (2) N(r){<=}A{sub 1}{gamma}(B{sub 1}r)/r, where N(r)=N(r,{lambda}{sub +}) or N(r)=N(r,{lambda}{sub -}), and |c{sub k}(r,v)|{<=}A{sub 2}{gamma}(B{sub 2}r). It is proved in addition that J{delta}({gamma})=JS({gamma})-JS({gamma}), where JS({gamma}) is the class of proper subharmonic functions of finite {gamma}-type.

Authors:
 [1]
  1. Ukrainian Academy of Banking, Sumy (Ukraine)
Publication Date:
OSTI Identifier:
21205612
Resource Type:
Journal Article
Resource Relation:
Journal Name: Sbornik. Mathematics; Journal Volume: 192; Journal Issue: 6; Other Information: DOI: 10.1070/SM2001v192n06ABEH000572; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPLEX MANIFOLDS; FUNCTIONS; MATHEMATICAL LOGIC

Citation Formats

Malyutin, K G. Fourier series and {delta}-subharmonic functions of finite {gamma}-type in a half-plane. United States: N. p., 2001. Web. doi:10.1070/SM2001V192N06ABEH000572; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Malyutin, K G. Fourier series and {delta}-subharmonic functions of finite {gamma}-type in a half-plane. United States. doi:10.1070/SM2001V192N06ABEH000572; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Malyutin, K G. Sat . "Fourier series and {delta}-subharmonic functions of finite {gamma}-type in a half-plane". United States. doi:10.1070/SM2001V192N06ABEH000572; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
@article{osti_21205612,
title = {Fourier series and {delta}-subharmonic functions of finite {gamma}-type in a half-plane},
author = {Malyutin, K G},
abstractNote = {Let {gamma}(r) be a growth function and let v(z) be a proper {delta}-subharmonic function in the sense of Grishin in a complex half-plane, that is v=v{sub 1}-v{sub 2}, where v{sub 1} and v{sub 2} are proper subharmonic functions (limsup{sub z{yields}}{sub t}v{sub i}(z){<=}0, for each real t, i=1,2), let {lambda}={lambda}{sub +}-{lambda}{sub -} be the full measure corresponding to v and let T(r,v) be its Nevanlinna characteristic. The class J{delta}({gamma}) of functions of finite {gamma}-type is defined as follows: v element of J{delta}({gamma}) if T(r,v){<=}A{gamma}(Br)/r for some positive constants A and B. The Fourier coefficients of v are defined in the standard way. The central result of the paper is the equivalence of the following properties: (1) v element of J{delta}({gamma}); (2) N(r){<=}A{sub 1}{gamma}(B{sub 1}r)/r, where N(r)=N(r,{lambda}{sub +}) or N(r)=N(r,{lambda}{sub -}), and |c{sub k}(r,v)|{<=}A{sub 2}{gamma}(B{sub 2}r). It is proved in addition that J{delta}({gamma})=JS({gamma})-JS({gamma}), where JS({gamma}) is the class of proper subharmonic functions of finite {gamma}-type.},
doi = {10.1070/SM2001V192N06ABEH000572; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},
journal = {Sbornik. Mathematics},
number = 6,
volume = 192,
place = {United States},
year = {Sat Jun 30 00:00:00 EDT 2001},
month = {Sat Jun 30 00:00:00 EDT 2001}
}
  • The paper deals with the space L{sup p(x)} consisting of classes of real measurable functions f(x) on [0,1] with finite integral ∫{sub 0}{sup 1}|f(x)|{sup p(x)} dx. If 1≤p(x)≤ p-bar <∞, then the space L{sup p(x)} can be made into a Banach space with the norm ∥f∥{sub p(⋅)}=inf(α > 0:∫{sub 0}{sup 1}|f(x)/α|{sup p(x)} dx≤ 1). The inequality ∥f−Q{sub n}(f)∥{sub p(⋅)}≤c(p)Ω(f,1/n){sub p(⋅)}, which is an analogue of the first Jackson theorem, is shown to hold for the finite Fourier-Haar series Q{sub n}(f), provided that the variable exponent p(x) satisfies the condition |p(x)−p(y)|ln (1/|x−y|)≤ c. Here, Ω(f,δ){sub p(⋅)} is the modulus of continuity in L{sup p(x)} defined inmore » terms of Steklov functions. If the function f(x) lies in the Sobolev space W{sub p(⋅)}{sup 1} with variable exponent p(x), it is shown that ∥f−Q{sub n}(f)∥{sub p(⋅)}≤c(p)/n∥f{sup ′}∥{sub p(⋅)}. Methods for estimating the deviation |f(x)−Q{sub n}(f,x)| for f(x)∈W{sub p(⋅)}{sup 1} at a given point x∈[0,1] are also examined. The value of sup{sub f∈W{sub p{sup 1}(1)}}|f(x)−Q{sub n}(f,x)| is calculated in the case when p(x)≡p= const, where W{sub p}{sup 1}(1)=(f∈W{sub p}{sup 1}:∥f{sup ′}∥{sub p(⋅)}≤1). Bibliography: 17 titles.« less
  • This report discusses subharmoic functions of finite order in a cone, also discussed are the weak convergence of the functions; the convergence of the functionals T{sub R}, conical-boundary density; the weak convergence on {Gamma}; conical-boundary steadiness; the growth regularity of functions with convergent functional Tr; and a regular distribution of the measure Tu, the growth regularity criteria.
  • Let v{sub n}(z) be a sequence of {delta}-subharmonic functions in some domain G. Conditions are studied under which the convergence of v{sub n}(z) as a sequence of generalized functions implies its convergence in the Lebesgue spaces L{sub p}({gamma}). Hoermander studied the case where v{sub n}(z) is a sequence of subharmonic functions and the measure {gamma} is the restriction of the Lebesgue measure to a compactum contained in G. In this paper a more general case is considered and theorems of two types are obtained. In theorems of the first type it is assumed that supp{gamma} subset of G. In theoremsmore » of the second type it is assumed that the support of the measure is a compactum and supp{gamma} subste of G-bar. In the second case, G is assumed to be the half-plane. Bibliography: 11 titles.« less
  • For an arbitrary open set {omega} subset of I{sup 2}=[0,1){sup 2} and an arbitrary function f element of L log {sup +}L log {sup +} log {sup +}L(I{sup 2}) such that f=0 on {omega} the double Fourier series of f with respect to the trigonometric system {psi}=E and the Walsh-Paley system {psi}=W is shown to converge to zero (over rectangles) almost everywhere on {omega}. Thus, it is proved that generalized localization almost everywhere holds on arbitrary open subsets of the square I{sup 2} for the double trigonometric Fourier series and the Walsh-Fourier series of functions in the class L logmore » {sup +}L log {sup +} log {sup +}L (in the case of summation over rectangles). It is also established that such localization breaks down on arbitrary sets that are not dense in I{sup 2}, in the classes {phi}{sub {psi}}(L)(I{sup 2}) for the orthonormal system {psi}=E and an arbitrary function such that {phi}{sub E}(u)=o(u log {sup +} log {sup +}u) as u{yields}{infinity} or for {phi}{sub W}(u)=u( log {sup +} log {sup +}u){sup 1-{epsilon}}, 0<{epsilon}<1.« less
  • Let f({xi},{eta}) be a function vanishing for {xi}{sup 2}+{eta}{sup 2}>r{sup 2}, where r is sufficiently small, and with Fourier series (of the function considered in the square (-{pi},{pi}]{sup 2}) or Fourier integral (of the function considered in the plane R{sup 2}) convergent uniformly or almost everywhere over rectangles. It is shown that a rotation of the system of coordinates through {pi}/4 {l_brace}{xi}=(x-y)/{radical}2, {eta}=(y+x)/{radical}2{r_brace} can 'damage' the convergence of the Fourier series or the Fourier integral of the resulting function.