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<p>We show that for any 1 ≤ <i>s</i> ≤ 2, there is a periodic continuous function <i>f</i> whose Fourier series is divergent at some point, and whose graph satisfies <Equation ID="Equ1"> <EquationSource Format="TEX">\(\text{dim}_{H}(\text{graph} \ f)=\text{dim}_{B}(\text{graph} \ f)=s.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mtext>dim</mtext> <mrow> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mtext>graph</mtext> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mtext>dim</mtext> <mrow> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mtext>graph</mtext> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo>.</mo> </math></EquationSource> </Equation></p>

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Fractal Dimensions and Divergence of Fourier Series

  • 期刊论文

  • 发表日期：

    2025年9月30日

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  • Mengjie Che,
  • Changhao Chen,
  • Jia Liu

摘要

We show that for any 1 ≤ s ≤ 2, there is a periodic continuous function f whose Fourier series is divergent at some point, and whose graph satisfies \(\text{dim}_{H}(\text{graph} \ f)=\text{dim}_{B}(\text{graph} \ f)=s.\) dim H ( graph f ) = dim B ( graph f ) = s .