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Abstract <p>A typical approximation problem is the interpolation problem. The classical method for solving it is to construct an interpolation polynomial. However, polynomials have a number of disadvantages, for instance, when used as a tool for approximating functions with singularities and functions with not very high smoothness. In practice, in order to approximate functions well, instead of constructing a high-degree interpolation polynomial, splines are used, which are very convenient to use. This paper examines the construction of interpolation splines using the Sobolev method, minimizing the norm in a certain Hilbert space. For the first time, Sobolev [<CitationRef CitationID="CR12">12</CitationRef>] posed the problem of finding the extremal function for the interpolation formula and calculating the norm of the error functional in the Sobolev space. In this work, the extremal function of the interpolation formula is found in explicit form in the Sobolev space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10489_Article_IEq3.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\(W_{2}^{{\left( m \right)}}({{R}^{n}})\)</EquationSource> <!--RusMath2570034Khayatov-m3--> </InlineEquation>; a function whose <i>m</i>th-order generalized derivatives are square integrable. Basically, the problem of constructing optimal interpolation formulas in the Sobolev space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10489_Article_IEq4.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tilde {W}_{2}^{{\left( m \right)}}\left( {{{T}_{1}}} \right)\)</EquationSource> <!--RusMath2570034Khayatov-m4--> </InlineEquation> for <i>m</i> = 4 is considered.</p>

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On Finding the Coefficients of the Optimal Interpolation Formula in the Sobolev Space \(\tilde {W}_{2}^{{\left( m \right)}}\)(T1)

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • Kh. U. Khayatov,
  • O. I. Jalolov

摘要

Abstract

A typical approximation problem is the interpolation problem. The classical method for solving it is to construct an interpolation polynomial. However, polynomials have a number of disadvantages, for instance, when used as a tool for approximating functions with singularities and functions with not very high smoothness. In practice, in order to approximate functions well, instead of constructing a high-degree interpolation polynomial, splines are used, which are very convenient to use. This paper examines the construction of interpolation splines using the Sobolev method, minimizing the norm in a certain Hilbert space. For the first time, Sobolev [12] posed the problem of finding the extremal function for the interpolation formula and calculating the norm of the error functional in the Sobolev space. In this work, the extremal function of the interpolation formula is found in explicit form in the Sobolev space \(W_{2}^{{\left( m \right)}}({{R}^{n}})\) ; a function whose mth-order generalized derivatives are square integrable. Basically, the problem of constructing optimal interpolation formulas in the Sobolev space \(\tilde {W}_{2}^{{\left( m \right)}}\left( {{{T}_{1}}} \right)\) for m = 4 is considered.