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<p>In this short paper we improve on the result from (Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, 1970) (Lemma 6.1) and based on that we show that any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation> dimensional subspace <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(V\)</EquationSource> </InlineEquation> of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C[a,b]\)</EquationSource> </InlineEquation> admits <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((n)\)</EquationSource> </InlineEquation>–alternating function <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation>, i.e., a non-zero function <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation>, for which there exists <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation> points <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(a\le t_1 &lt; \ldots &lt; t_{{n}}\le b\)</EquationSource> </InlineEquation> such that <Equation ID="Equa"> <EquationSource Format="TEX">\(\begin{aligned}&amp;|f(t_j)| =||f||,&amp;\text{for }j=1,\ldots,n, \\&amp;f(t_j) = -f(t_{j+1}),&amp;\text{for }j=1,\ldots,n-1.\end{aligned}\)</EquationSource> </Equation></p>

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On alternating functions in subspaces of \(C[a,b]\)

  • 期刊论文

  • 发表日期：

    2025年7月9日

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  • Boris Shekhtman,
  • Lesław Skrzypek

摘要

In this short paper we improve on the result from (Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, 1970) (Lemma 6.1) and based on that we show that any \(n\) dimensional subspace \(V\) of \(C[a,b]\) admits \((n)\) –alternating function \(f\) , i.e., a non-zero function \(f\) , for which there exists \(n\) points \(a\le t_1 < \ldots < t_{{n}}\le b\) such that \(\begin{aligned}&|f(t_j)| =||f||,&\text{for }j=1,\ldots,n, \\&f(t_j) = -f(t_{j+1}),&\text{for }j=1,\ldots,n-1.\end{aligned}\)