Scholarlink

Abstract <p> Suppose <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f\in L^{1}[-\pi, \pi]\)</EquationSource> </InlineEquation> has a Fourier series <Equation ID="Equi"> <EquationSource Format="TEX">\(\sum_{k=-\infty}^{\infty} \widehat{f}(n_{k})e^{in_{k}x} \quad (n_{-k}=-n_{k})\)</EquationSource> </Equation> with small gaps <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n_{k+1}-n_{k}\geq q &gt;1\)</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(k\geq0\)</EquationSource> </InlineEquation>. Here, by applying the Wiener–Ingham result for finite trigonometric sum with small gaps, we obtain a sufficient condition for the convergence of the series <Equation ID="Equii"> <EquationSource Format="TEX">\(\sum_{k\in \mathbb{Z}}|\widehat{f}(n_{k})|^{\beta}\quad (0&lt;\beta\leq2)\)</EquationSource> </Equation> if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation> is locally of the class <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\Lambda BV^{(p)}\)</EquationSource> </InlineEquation>. </p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On \(\beta\)-Absolute Convergence of Fourier Series with Small Gaps

  • 期刊论文

  • 发表日期：

    2025年8月1日

查看全文
  • K. N. Darji

摘要

Abstract

Suppose \(f\in L^{1}[-\pi, \pi]\) has a Fourier series \(\sum_{k=-\infty}^{\infty} \widehat{f}(n_{k})e^{in_{k}x} \quad (n_{-k}=-n_{k})\) with small gaps \(n_{k+1}-n_{k}\geq q >1\) for all \(k\geq0\) . Here, by applying the Wiener–Ingham result for finite trigonometric sum with small gaps, we obtain a sufficient condition for the convergence of the series \(\sum_{k\in \mathbb{Z}}|\widehat{f}(n_{k})|^{\beta}\quad (0<\beta\leq2)\) if \(f\) is locally of the class \(\Lambda BV^{(p)}\) .