Email this article Generalized Localization for Spherical Partial Sums of Multiple Fourier Series
- Authors: Ashurov R.R.1,2
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Affiliations:
- National University of Uzbekistan Named after Mirzo Ulugbek
- Romanovskii Uzbekistan Academy of Science Institute of Mathematics, Uzbekistan Academy of Science
- Issue: Vol 100, No 3 (2019)
- Pages: 505-507
- Section: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/225732
- DOI: https://doi.org/10.1134/S1064562419060012
- ID: 225732
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Abstract
Abstract—In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2 class is proved, that is, if f ∈ L2(TN) and f = 0 on an open set Ω ⊂ TN, then it is shown that the spherical partial sums of this function converge to zero almost-everywhere on Ω. It has been previously known that the generalized localization is not valid in Lp(TN) when \(1 \leqslant p < 2\). Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp(TN), p ≥ 1: if p ≥ 2 then we have the generalized localization and if p < 2, then the generalized localization fails.
About the authors
R. R. Ashurov
National University of Uzbekistan Named after Mirzo Ulugbek; Romanovskii Uzbekistan Academy of Science Institute of Mathematics, Uzbekistan Academy of Science
Author for correspondence.
Email: ashurovr@gmail.com
Uzbekistan, Tashkent, 100170; Tashkent, 100170
