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<p>In the paper, we give necessary and sufficient conditions for a continuous on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\([-\pi ,\pi ]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> function <i>f</i> to belong various generalized Lipschitz classes defined by the Jacobi-Dunkl translation in terms of Fourier-Jacobi-Dunkl coefficients. As a corollary, we obtain analogues of Boas equivalence results and their extensions due to Tikhonov and Moricz. Also, we give sufficient conditions for generalized absolute convergence of Fourier-Jacobi-Dunkl series and show its sharpness in important <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> case using a new variant of inverse approximation theorem.</p>

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Generalized absolute convergence of Jacobi-Dunkl series and Lipschitz classes in uniform and integral metrics

  • 期刊论文

  • 发表日期：

    2026年1月17日

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  • Sergey Volosivets

摘要

In the paper, we give necessary and sufficient conditions for a continuous on \([-\pi ,\pi ]\) [ - π , π ] function f to belong various generalized Lipschitz classes defined by the Jacobi-Dunkl translation in terms of Fourier-Jacobi-Dunkl coefficients. As a corollary, we obtain analogues of Boas equivalence results and their extensions due to Tikhonov and Moricz. Also, we give sufficient conditions for generalized absolute convergence of Fourier-Jacobi-Dunkl series and show its sharpness in important \(L^2\) L 2 case using a new variant of inverse approximation theorem.