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<p>Gegenbauer polynomial approximations play an important role in applied mathematics and computational physics. In this paper, we present sharp bounds for Gegenbauer expansion coefficients of functions belonging to fractional spaces and then derive some new and sharp error bounds for Gegenbauer approximations in weighted <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2759_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2759_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norms. Ample numerical results are provided to demonstrate the sharpness of the estimates.</p>

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Optimal Error Estimates for Gegenbauer Approximations in Fractional Spaces

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Ruiyi Xie,
  • Wenjie Liu,
  • Haiyong Wang,
  • Boying Wu

摘要

Gegenbauer polynomial approximations play an important role in applied mathematics and computational physics. In this paper, we present sharp bounds for Gegenbauer expansion coefficients of functions belonging to fractional spaces and then derive some new and sharp error bounds for Gegenbauer approximations in weighted \(L^{\infty }\) L ∞ and \(L^2\) L 2 norms. Ample numerical results are provided to demonstrate the sharpness of the estimates.