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Abstract <p>In this work some uniqueness theorems for series with respect to the bounded Ciesielski system are proved. In particular, if the partial sums <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S_{l}(x)=\sum_{n=-k+2}^{l}a_{n}F_{n}(x)\)</EquationSource> <!--ContMath2670010Gevorkyan-m1--> </InlineEquation> of the bounded Ciesielski series <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sum_{n=-k+2}^{\infty}a_{n}F_{n}(x)\)</EquationSource> <!--ContMath2670010Gevorkyan-m2--> </InlineEquation> converge in measure to a bounded function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(f\)</EquationSource> <!--ContMath2670010Gevorkyan-m3--> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sup_{l}|S_{l}(x)|&lt;\infty\)</EquationSource> <!--ContMath2670010Gevorkyan-m4--> </InlineEquation> when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(x\notin B\)</EquationSource> <!--ContMath2670010Gevorkyan-m5--> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(B\)</EquationSource> <!--ContMath2670010Gevorkyan-m6--> </InlineEquation> is some countable set, then this series is the Fourier series of the function <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(f\)</EquationSource> <!--ContMath2670010Gevorkyan-m7--> </InlineEquation> with respect to the bounded Ciesielski system.</p>

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Uniqueness of Series with Respect to the Bounded Ciesielski System

  • 期刊论文

  • 发表日期：

    2026年6月20日

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  • G. Gevorkyan,
  • K. Keryan,
  • M. Poghosyan

摘要

Abstract

In this work some uniqueness theorems for series with respect to the bounded Ciesielski system are proved. In particular, if the partial sums \(S_{l}(x)=\sum_{n=-k+2}^{l}a_{n}F_{n}(x)\) of the bounded Ciesielski series \(\sum_{n=-k+2}^{\infty}a_{n}F_{n}(x)\) converge in measure to a bounded function \(f\) and \(\sup_{l}|S_{l}(x)|<\infty\) when \(x\notin B\) , where \(B\) is some countable set, then this series is the Fourier series of the function \(f\) with respect to the bounded Ciesielski system.