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Abstract <p> We prove the existence of a sequence of positive integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n_i\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(i\in\mathbb{N}\)</EquationSource> </InlineEquation>, that has zero density in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb{N}\)</EquationSource> </InlineEquation> and possesses the following property: if the subsequence <Equation ID="Equi"> <EquationSource Format="TEX">\(S_{n_i}(x)=\sum_{k=1}^{n_i}a_k\chi_k(x)\)</EquationSource> </Equation> of partial sums of a Haar series converges everywhere to an everywhere finite integrable function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation>, then this series is the Fourier–Haar series of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation>. </p>

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On the Uniqueness of Haar Series Converging over Subsequences of Partial Sums

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • G. G. Gevorkyan

摘要

Abstract

We prove the existence of a sequence of positive integers \(n_i\) , \(i\in\mathbb{N}\) , that has zero density in \(\mathbb{N}\) and possesses the following property: if the subsequence \(S_{n_i}(x)=\sum_{k=1}^{n_i}a_k\chi_k(x)\) of partial sums of a Haar series converges everywhere to an everywhere finite integrable function \(f\) , then this series is the Fourier–Haar series of \(f\) .