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<p>This paper introduces a novel algorithm for approximating bivariate functions, addressing both smooth and non-smooth cases efficiently. We extend the concept of domain splitting to two dimensions, presenting a Padé-based nonlinear approximation method using Chebyshev series coefficients. Our approach approximates smooth functions to machine precision and ensures non-oscillatory profiles for non-smooth functions. Notably, we present the first theoretical convergence results using approximated coefficients, departing from existing literature that relies on exact Chebyshev or series coefficients. We provide a comprehensive <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10543_2025_1059_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-convergence analysis for functions of bounded variation in the sense of Hardy on rectangular domains. The method incorporates quadrature formulas for practical coefficient approximation, significantly enhancing its real-world applicability. A key advantage is its independence from prior knowledge of singularity locations, although currently limited to functions with discontinuities along coordinate axes. Numerical results demonstrate the algorithm’s effectiveness in mitigating the Gibbs phenomenon for non-smooth functions with finite Vitali variation, reducing it to negligible levels.</p>

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Padé-based nonlinear approximation of bivariate non-smooth functions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月21日

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  • S. Akansha

摘要

This paper introduces a novel algorithm for approximating bivariate functions, addressing both smooth and non-smooth cases efficiently. We extend the concept of domain splitting to two dimensions, presenting a Padé-based nonlinear approximation method using Chebyshev series coefficients. Our approach approximates smooth functions to machine precision and ensures non-oscillatory profiles for non-smooth functions. Notably, we present the first theoretical convergence results using approximated coefficients, departing from existing literature that relies on exact Chebyshev or series coefficients. We provide a comprehensive \(L^1\) L 1 -convergence analysis for functions of bounded variation in the sense of Hardy on rectangular domains. The method incorporates quadrature formulas for practical coefficient approximation, significantly enhancing its real-world applicability. A key advantage is its independence from prior knowledge of singularity locations, although currently limited to functions with discontinuities along coordinate axes. Numerical results demonstrate the algorithm’s effectiveness in mitigating the Gibbs phenomenon for non-smooth functions with finite Vitali variation, reducing it to negligible levels.