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Generalized Localization and Summability Almost Everywhere of Multiple Fourier Series and Integrals

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Abstract

It is well known that Luzin’s conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzin’s conjecture has been achieved by considering simpler problems. In this paper, we consider three of these problems for spherical partial sums: the principle of generalized localization, summability almost everywhere, and convergence almost everywhere of multiple Fourier series of smooth functions. A brief overview of the work in these areas is given and unsolved problems are mentioned and new problems are formulated. Moreover, at the end of the work, a new result on the convergence of spherical sums for functions from Sobolev classes is proved.

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Authors and Affiliations

  1. National University of Uzbekistan named after M. Ulugbek, Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

    R. R. Ashurov

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Correspondence to R. R. Ashurov.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 67, No. 4, Science — Technology — Education — Mathematics — Medicine, 2022.

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Ashurov, R.R. Generalized Localization and Summability Almost Everywhere of Multiple Fourier Series and Integrals. J Math Sci 278, 408–425 (2024). https://doi.org/10.1007/s10958-024-06930-7

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