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Cesàro summability and Lebesgue points of higher dimensional Fourier series

  • Corresponding author: Ferenc Weisz

    Corresponding author: Ferenc Weisz

This research was supported by the Hungarian Scientific Research Funds (OTKA) No KH130426

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  • We give four generalizations of the classical Lebesgue's theorem to multi-dimensional functions and Fourier series. We introduce four new concepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, we prove that the Cesàro means $ \sigma_n^{\alpha}f $ of the Fourier series of a multi-dimensional function converge to $ f $ at each Lebesgue point as $ n\to \infty $.

    Mathematics Subject Classification: Primary: 42B08; Secondary 42A24, 42B25.

    Citation:

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  • Figure 1.  Regions of the $ \ell_q $-partial sums for $ d = 2 $

    Figure 2.  The cone for $ d = 2 $

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