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Abstract <p>The problem of finding the exact upper bound on the values of the moduli of continuity of conjugate functions with a fixed step value <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11970_2025_7227_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(h\)</EquationSource> <!--BMatMGU2570051Popov-m3--> </InlineEquation> is studied. The exact upper bound is taken from a set of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11970_2025_7227_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(2\pi\)</EquationSource> <!--BMatMGU2570051Popov-m4--> </InlineEquation>-periodic functions that satisfy the Lipschitz condition of the first order with a given Lipschitz constant. Extreme asymptotics in this problem are found with an accuracy up to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11970_2025_7227_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(h^{3})\)</EquationSource> <!--BMatMGU2570051Popov-m5--> </InlineEquation> for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11970_2025_7227_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(h\rightarrow 0+\)</EquationSource> <!--BMatMGU2570051Popov-m6--> </InlineEquation>. The margin of error is estimated clearly.</p>

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Optimal Estimate of the Modulus of Continuity of a Function Conjugate to a \(\boldsymbol{2\pi}\)-Periodic Lipschitz One

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • A. Yu. Popov,
  • V. A. Okulov

摘要

Abstract

The problem of finding the exact upper bound on the values of the moduli of continuity of conjugate functions with a fixed step value \(h\) is studied. The exact upper bound is taken from a set of \(2\pi\) -periodic functions that satisfy the Lipschitz condition of the first order with a given Lipschitz constant. Extreme asymptotics in this problem are found with an accuracy up to \(O(h^{3})\) for \(h\rightarrow 0+\) . The margin of error is estimated clearly.