Open AccessGeneralized Localization and Summability Almost Everywhere of Multiple Fourier Series and Integrals
Author(s) -
Ravshan Ashurov
Publication year - 2021
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2021-67-4-634-653
Subject(s) - almost everywhere , mathematics , fourier series , function series , series (stratigraphy) , conjecture , trigonometric series , fourier inversion theorem , convergence (economics) , fourier transform , sobolev space , mathematical analysis , trigonometry , fourier analysis , pure mathematics , fractional fourier transform , paleontology , economics , biology , economic growth
It is well known that Luzins 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 Luzins hypothesis 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.