Moment problems for bounded functions
type: Item
instance of: scholarly article
title: Moment problems for bounded functions
zbMATH Open document ID: 0836.41005
author: Konstantin M. Dyakonov
published in: Communications in Analysis and Geometry
publication date: 1996-05-09T00:00:00Z
review text: Let \((X,\mu)\) be a measure space and \(\{ \varphi_j\}\) be a sequence of functions in \(L^1 (X, \mu)\) normalized by \[ |\varphi_j |_j= \int|\varphi_j |d\mu=1. \] The main result of the paper describes a sufficient condition which guarantees that for any sequence \(\{a_j \}\in l^\infty\) there exists \(f\in L^\infty (X, \mu)\) for which \[ \int f\varphi_j d\mu= a_j, \qquad j=1, 2,\dots\;.\tag{1} \] The condition says: there is a collection \(\{E_j \}^\infty_{j =1}\) of pairwise disjoint measurable sets \(E_j \subset X\) such that \[ \sigma= \inf_j \int_{E_j} |\varphi_j |d\mu> 1/2. \] Moreover the solution \(f\) of (1) satisfies estimation \[ |f|_\infty\leq (2\sigma- 1)\sup_j |a_j|. \] The paper also contains several interesting applications of this abstract theorem to interpolation problems for bounded harmonic functions in \(R_+^{n+1}\), to estimates of the growth of entire functions represented by lacunary series and to interpolation problems for BMOA class in the unit disc. These applications and other ones presented in the work illustrate the power of general interpolation theorem.
reviewed by: Jan Janas
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 827485
zbMATH Keywords: interpolation by harmonic functions
zbMATH Keywords: Fourier series
zbMATH Keywords: BMOA
MaRDI profile type: Publication
full work available at URL: cag.1994.v2.n4.a2
OpenAlex ID: W2740576003
DOI: 10.4310/CAG.1994.V2.N4.A2
description: scientific article; zbMATH DE number 827485