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<p>We study a system of polynomials orthonormal with respect to a Sobolev-type inner product and associated with classical Hermite polynomials.It is shown that for functions from a&#xa0;weighted Sobolev space the Fourier series in this system converges uniformly on an interval provided that the parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1597_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ p $</EquationSource> </InlineEquation> is not less than two.For the cases where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1597_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ p $</EquationSource> </InlineEquation> is less than two, we construct an example of&#xa0;a&#xa0;function whose Fourier series diverges at a given point.We also investigate the question of absolute convergence of the Fourier series in this system of polynomials on an interval.</p>

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Uniform and Absolute Convergence of the Fourier Series in Hermite–Sobolev Polynomials

  • 期刊论文

  • 发表日期：

    2025年9月30日

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  • R. M. Gadzhimirzaev

摘要

We study a system of polynomials orthonormal with respect to a Sobolev-type inner product and associated with classical Hermite polynomials.It is shown that for functions from a weighted Sobolev space the Fourier series in this system converges uniformly on an interval provided that the parameter $ p $ is not less than two.For the cases where $ p $ is less than two, we construct an example of a function whose Fourier series diverges at a given point.We also investigate the question of absolute convergence of the Fourier series in this system of polynomials on an interval.