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Abstract <p> A solution is given to Stechkin’s problem on the best approximation, in the uniform norm on the real axis, of differentiation operators of fractional (more precisely, real) order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation> by bounded linear operators from the space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L_2\)</EquationSource> </InlineEquation> to the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C\)</EquationSource> </InlineEquation> on the class <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal{Q}^n\)</EquationSource> </InlineEquation> of functions whose <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>th-order fractional derivative, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(0\le k&lt;n,\)</EquationSource> </InlineEquation> has a summable Fourier transform. The corresponding sharp Kolmogorov inequality is presented. A solution is obtained to the problem of optimal recovery of the differentiation operator of fractional order <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation> on functions of the class <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathcal{Q}^n\)</EquationSource> </InlineEquation> defined with a known error in the space <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(L_2\)</EquationSource> </InlineEquation>. </p>

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Best Approximation of a Fractional-Order Differentiation Operator in the Uniform Norm on the Axis on the Class of Functions with Summable Fourier Transform of the Highest Derivative

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • V. V. Arestov

摘要

Abstract

A solution is given to Stechkin’s problem on the best approximation, in the uniform norm on the real axis, of differentiation operators of fractional (more precisely, real) order \(k\) by bounded linear operators from the space \(L_2\) to the space \(C\) on the class \(\mathcal{Q}^n\) of functions whose \(n\) th-order fractional derivative, \(0\le k<n,\) has a summable Fourier transform. The corresponding sharp Kolmogorov inequality is presented. A solution is obtained to the problem of optimal recovery of the differentiation operator of fractional order \(k\) on functions of the class \(\mathcal{Q}^n\) defined with a known error in the space \(L_2\) .