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Abstract <p> We describe the set of parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((p_1,p_2,q_1,q_2)\)</EquationSource> </InlineEquation> such that the balls <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(B_{q_1,q_2}^{s,b}\)</EquationSource> </InlineEquation> are rigid in the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\ell_{q_1,q_2}^{s,b}\)</EquationSource> </InlineEquation> metric; i.e., they are poorly approximated by linear subspaces of dimension <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\le}\,(1-\varepsilon)sb\)</EquationSource> </InlineEquation> for large <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(s\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(b\)</EquationSource> </InlineEquation>. Thus we have settled an important qualitative case in the problem of estimating the widths of balls in mixed norms. The proof combines lower bounds from our previous papers and a new construction for the approximation by linear subspaces in the so-called exceptional case. </p>

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Kolmogorov Widths of Balls in Mixed Norms: The Case of Rigidity

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Yuri Malykhin,
  • Konstantin Ryutin

摘要

Abstract

We describe the set of parameters \((p_1,p_2,q_1,q_2)\) such that the balls \(B_{q_1,q_2}^{s,b}\) are rigid in the \(\ell_{q_1,q_2}^{s,b}\) metric; i.e., they are poorly approximated by linear subspaces of dimension \({\le}\,(1-\varepsilon)sb\) for large \(s\) and \(b\) . Thus we have settled an important qualitative case in the problem of estimating the widths of balls in mixed norms. The proof combines lower bounds from our previous papers and a new construction for the approximation by linear subspaces in the so-called exceptional case.