<p>We introduce a new concept of Lebesgue points for higher dimensionalfunctions. Every continuity point is a Lebesgue point and almost everypoint is a Lebesgue point of an integrable function. Given a strictly increasingcontinuous function<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1504_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>, we prove that the Fejér or Cesàro means<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1504_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma_n^{\alpha}f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>σ</mi> <mi>n</mi> <mi>α</mi> </msubsup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> of the Fourierseries of a two-dimensional function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1504_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in L_1(\mathbb{T}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> converge to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1504_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$f$</EquationSource> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> </InlineEquation> at each Lebesguepoint as <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1504_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$n\to \infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and n is in the cone around the graph of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1504_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>. We also prove thisresult for higher dimensional functions and for other summability means. This isa generalization of the classical one-dimensional Lebesgue’s theorem for the Fejérmeans.</p>

Convergence of summability means of higher dimensional Fourier series and Lebesgue points

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2025年2月7日

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F. Weisz

摘要

We introduce a new concept of Lebesgue points for higher dimensionalfunctions. Every continuity point is a Lebesgue point and almost everypoint is a Lebesgue point of an integrable function. Given a strictly increasingcontinuous function $\delta$ $δ$ , we prove that the Fejér or Cesàro means $\sigma_n^{\alpha}f$ $σ_{n}^{α} f$ of the Fourierseries of a two-dimensional function $f\in L_1(\mathbb{T}^2)$ $f \in L_{1} (T^{2})$ converge to $f$ $f$ at each Lebesguepoint as $n\to \infty$ $n \to \infty$ and n is in the cone around the graph of $\delta$ $δ$ . We also prove thisresult for higher dimensional functions and for other summability means. This isa generalization of the classical one-dimensional Lebesgue’s theorem for the Fejérmeans.