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Abstract <p>The paper deals with the convergence of the special series of moduli of difference of general Fourier coefficients of functions from some differentiable class, with respect to general orthonormal systems (ONSs). It has been shown that the functions with derivatives from the class <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7234_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{Lip}\ 1\)</EquationSource> <!--ContMath2460173Tsagareishvili-m1--> </InlineEquation> on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7234_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\([0,1]\)</EquationSource> <!--ContMath2460173Tsagareishvili-m2--> </InlineEquation> have convergent series of that kind with respect to Haar or trigonometric systems (see [<CitationRef CitationID="CR1">1</CitationRef>, Ch. <CitationRef CitationID="CR2">2</CitationRef>]). We have proven that with respect to general ONSs this series, in general, is not convergent. For this reason, it becomes necessary to find a condition that must be imposed on the functions of ONS so that this special series converges for functions with derivatives from the class <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7234_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{Lip}\ 1\)</EquationSource> <!--ContMath2460173Tsagareishvili-m3--> </InlineEquation>. We have also shown that the obtained results are the best possible.</p>

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Properties of General Fourier Coefficients of the Smooth Functions

  • 期刊论文

  • 发表日期：

    2025年8月26日

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  • V. Tsagareishvili,
  • A. Kashibadze

摘要

Abstract

The paper deals with the convergence of the special series of moduli of difference of general Fourier coefficients of functions from some differentiable class, with respect to general orthonormal systems (ONSs). It has been shown that the functions with derivatives from the class \(\textrm{Lip}\ 1\) on \([0,1]\) have convergent series of that kind with respect to Haar or trigonometric systems (see [1, Ch. 2]). We have proven that with respect to general ONSs this series, in general, is not convergent. For this reason, it becomes necessary to find a condition that must be imposed on the functions of ONS so that this special series converges for functions with derivatives from the class \(\textrm{Lip}\ 1\) . We have also shown that the obtained results are the best possible.