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Abstract <p> In the Bergman space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(B_{q,\gamma}\)</EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((1\leq q&lt;\infty\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\gamma:=\gamma(|z|)&gt; 0)\)</EquationSource> </InlineEquation>, we find sharp inequalities between the best simultaneous approximation of a function and the averaged moduli of smoothness <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\omega_2(f^{(r)},t)_{H_{q,R}}\)</EquationSource> </InlineEquation> of the angular boundary values of the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(r\)</EquationSource> </InlineEquation>th derivatives. These inequalities are applied to the problem of evaluation of the supremum of best simultaneous approximations of some classes of functions defined in terms of moduli of smoothness and lying in the Bergman space <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(B_{q,\gamma}\)</EquationSource> </InlineEquation>. </p>

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On Best Simultaneous Approximation of Analytic Functions in the Weighted Bergman Space

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • M. Sh. Shabozov,
  • D. K. Tukhliev

摘要

Abstract

In the Bergman space \(B_{q,\gamma}\) \((1\leq q<\infty\) , \(\gamma:=\gamma(|z|)> 0)\) , we find sharp inequalities between the best simultaneous approximation of a function and the averaged moduli of smoothness \(\omega_2(f^{(r)},t)_{H_{q,R}}\) of the angular boundary values of the \(r\) th derivatives. These inequalities are applied to the problem of evaluation of the supremum of best simultaneous approximations of some classes of functions defined in terms of moduli of smoothness and lying in the Bergman space \(B_{q,\gamma}\) .