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Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 3174–3183 | Cite as

Generalized Localization for Spherical Partial Sums of Multiple Fourier Series

  • Ravshan AshurovEmail author
Article
  • 19 Downloads

Abstract

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the \(L_2\)-class is proved, that is, if \(f\in L_2({\mathbb {T}}^N)\) and \(f=0\) on an open set \(\Omega \subset {\mathbb {T}}^N\), then it is shown that the spherical partial sums of this function converge to zero almost-everywhere on \(\Omega \). It has been previously known that the generalized localization is not valid in \(L_p({\mathbb {T}}^N)\) when \(1\le p<2\). Thus the problem of generalized localization for the spherical partial sums is completely solved in \(L_p({\mathbb {T}}^N)\), \(p\ge 1\): if \(p\ge 2\) then we have the generalized localization and if \(p<2\), then the generalized localization fails.

Keywords

Multiple Fourier series Spherical partial sums Convergence almost-everywhere Generalized localization 

Mathematics Subject Classification

Primary 42B05 Secondary 42B99 

Notes

Acknowledgements

The author conveys thanks to Sh. A. Alimov for discussions of this result and gratefully acknowledges Marcelo M. Disconzi (Vanderbilt University, USA) for support and hospitality. I would also like to express my special thanks to JFAA’s anonymous reviewers for their remarks, which considerably improved the content of this paper. The author was supported by Foundation for Support of Basic Research of the Republic of Uzbekistan (Project Number is OT-F4-88).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National University of Uzbekistan Named After Mirzo UlugbekTashkentUzbekistan
  2. 2.Institute of MathematicsUzbekistan Academy of ScienceTashkentUzbekistan

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