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Abstract <p> The paper presents sufficient conditions for the generalized absolute convergence of series of Fourier–Jacobi coefficients (with factors) of functions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p[0,\pi]\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(1&lt;p\leq 2\)</EquationSource> </InlineEquation>, with Jacobi weight, where the factors satisfy the reverse Hölder inequality. A proof is given that this result cannot be improved in the case of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p=2\)</EquationSource> </InlineEquation> under certain restrictions. An analog of the Titchmarsh equivalence theorem on the relationship between the smoothness of a function and the behavior of the remainder of a series of its Fourier–Jacobi coefficients with a power weight is also given. </p>

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Generalized Absolute Convergence of Fourier–Jacobi Series

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • S. S. Volosivets

摘要

Abstract

The paper presents sufficient conditions for the generalized absolute convergence of series of Fourier–Jacobi coefficients (with factors) of functions in \(L^p[0,\pi]\) , \(1<p\leq 2\) , with Jacobi weight, where the factors satisfy the reverse Hölder inequality. A proof is given that this result cannot be improved in the case of \(p=2\) under certain restrictions. An analog of the Titchmarsh equivalence theorem on the relationship between the smoothness of a function and the behavior of the remainder of a series of its Fourier–Jacobi coefficients with a power weight is also given.