On convergence in the mean
type: Item
instance of: scholarly article
title: On convergence in the mean
zbMATH Open document ID: 0857.30002
author: G. S. Kocharian
published in: Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences
publication date: 1998-06-24T00:00:00Z
review text: Let \(G\) be an arbitrary bounded simply connected domain in \(\mathbb{C}\) with a rectifiable Jordan boundary \(\Gamma\). A function \(f\) holomorphic on \(G\) is said to be of the class \(E_p\) if there exists a constant \(K>0\) such that \(\int_{\Gamma_r} |f(z)|^p |dz|\leq K\) for any closed, rectifiable curve \(\Gamma_r \subset G\). It is shown that for domains \(G\) with a special boundary curve \(\Gamma\) the functions \(f\) of class \(E_p\) can be expanded into series of a certain Walsh system \((M_k)_k\) of rational functions with given sets of poles with convergence in the mean: \[ \lim_{n\to\infty} \int_\Gamma \Bigl|f(z)-\sum^n_{k=1} a_kM_k(z) \Bigr|^p |dz |=0. \]
Mathematics Subject Classification ID: 30B60
zbMATH DE Number: 907156
zbMATH Keywords: holomorphic function
zbMATH Keywords: series expansion
zbMATH Keywords: Walsh system
zbMATH Keywords: convergence in the mean
reviewed by: Fritz Haslinger
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 907156