On the Poisson representation of a function harmonic in the upper half-plane
type: Item
instance of: scholarly article
title: On the Poisson representation of a function harmonic in the upper half-plane
zbMATH Open document ID: 1034.31001
DOI: 10.1007/BF03321016
author: Seçil Gergün
author: I. V. Ostrovskii
published in: Computational Methods and Function Theory
publication date: 2003-10-22T00:00:00Z
review text: The authors discuss conditions for the validity of the generalized Poisson formula for a function \(u(z)\) harmonic in the half-plane \(\operatorname{Im} z > 0\) which has the form \(u(z)= \int_{-\infty}^{+\infty} P_q (z,t) \,d \nu (t)+ \operatorname{Im} P(z)\), where \(\nu\) is a \(\sigma\)-finite Borel measure, \(P\) is a real polynomial, \(P_q(z,t)= \operatorname{Im} [(1+tz)^q \pi^{-1} (t-z)^{-1} (1+t^2)^{-q}]\). These conditions differ from known ones by weaker growth restrictions in \(\operatorname{Im} z > 0\) and stronger behavior on the real axis \(\operatorname{Im} z = 0\).
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 31A10
zbMATH DE Number: 1995501
zbMATH Keywords: Poisson integral
zbMATH Keywords: Nevanlinna formula
zbMATH Keywords: Phragmén-Lindelöf principle
reviewed by: Vladimir Mityushev
MaRDI profile type: Publication
cites work: On the Titchmarsh convolution theorem
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description: scientific article; zbMATH DE number 1995501