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Abstract <p> In the present paper we discuss the rate of the approximation in norm by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1123_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(d\)</EquationSource> </InlineEquation>-dimensional Marcinkiewicz-type matrix transform means of Walsh-Fourier series for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1123_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^p(G^{d})\)</EquationSource> </InlineEquation> (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1123_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\leq p &lt;\infty\)</EquationSource> </InlineEquation>) and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1123_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(C(G^{d})\)</EquationSource> </InlineEquation> functions. Matrix transform means are generalizations of many well-known summation methods, such as Fejér (or <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1123_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\((C,1)\)</EquationSource> </InlineEquation>), Cesàro (or <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1123_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\((C,\alpha)\)</EquationSource> </InlineEquation>), Riesz, Nörlund and weighted means. </p>

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Approximation by \(d\)-Dimensional Marcinkiewicz Type Matrix Transform Means of Walsh-Fourier Series

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • István Blahota

摘要

Abstract

In the present paper we discuss the rate of the approximation in norm by \(d\) -dimensional Marcinkiewicz-type matrix transform means of Walsh-Fourier series for \(L^p(G^{d})\) ( \(1\leq p <\infty\) ) and \(C(G^{d})\) functions. Matrix transform means are generalizations of many well-known summation methods, such as Fejér (or \((C,1)\) ), Cesàro (or \((C,\alpha)\) ), Riesz, Nörlund and weighted means.