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 Mat. Zametki, 1998, Volume 64, Issue 1, Pages 24–36 (Mi mz1369)

Divergence almost everywhere of rectangular partial sums of multiple Fourier series of bounded functions

S. Galstyana, G. A. Karagulianb

a Yerevan State University
b Institute of Mathematics, National Academy of Sciences of Armenia

Abstract: In this paper we establish the following results, which are the multidimensional generalizations of well-known theorems:
• 1) Suppose that a function $f\in C(\mathbb T^m)$ has no intervals of constancy in $\mathbb T^m$; then there exists a homeomorphism $\varphi\colon\mathbb T^m\to\mathbb T^m$ such that the Fourier series of the superposition $F=f\circ\varphi$ is divergent with respect to rectangles almost everywhere;
• 2) for any integrable function $f\in L^1(\mathbb T^m)$, with $|f(\mathbf x)|\geqslant\alpha>0$, $x\in\mathbb T^m$, there exists a signum function $\varepsilon(\mathbf x)=\pm 1$, $\mathbf x\in\mathbb T^m$ such that the Fourier series of the product $f(\mathbf x)\varepsilon(\mathbf x)$ is divergent with respect to rectangles almost everywhere.

DOI: https://doi.org/10.4213/mzm1369

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English version:
Mathematical Notes, 1998, 64:1, 20–30

Bibliographic databases:

UDC: 517

Citation: S. Galstyan, G. A. Karagulian, “Divergence almost everywhere of rectangular partial sums of multiple Fourier series of bounded functions”, Mat. Zametki, 64:1 (1998), 24–36; Math. Notes, 64:1 (1998), 20–30

Citation in format AMSBIB
\Bibitem{GalKar98} \by S.~Galstyan, G.~A.~Karagulian \paper Divergence almost everywhere of rectangular partial sums of multiple Fourier series of bounded functions \jour Mat. Zametki \yr 1998 \vol 64 \issue 1 \pages 24--36 \mathnet{http://mi.mathnet.ru/mz1369} \crossref{https://doi.org/10.4213/mzm1369} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1694010} \zmath{https://zbmath.org/?q=an:0918.42005} \transl \jour Math. Notes \yr 1998 \vol 64 \issue 1 \pages 20--30 \crossref{https://doi.org/10.1007/BF02307192} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000078147600004} 

• http://mi.mathnet.ru/eng/mz1369
• https://doi.org/10.4213/mzm1369
• http://mi.mathnet.ru/eng/mz/v64/i1/p24

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This publication is cited in the following articles:
1. I. L. Bloshanskii, “A Criterion for Weak Generalized Localization in the Class $L_1$ for Multiple Trigonometric Series from the Viewpoint of Isometric Transformations”, Math. Notes, 71:4 (2002), 464–476
2. Bloshanskii I., “Linear Transformations of R-N and Problems of Convergence of Multiple Fourier Integral”, Wavelet Analysis and Active Media Technology Vols 1-3, ed. Li J. Jaffard S. Suen C. Daugman J. Wickerhauser V. Torresani B. Yen J. Zhong N. Pal S., World Scientific Publ Co Pte Ltd, 2005, 1081–1091
3. Bloshanskii I.L., “Linear Transformations of R-N and Problems of Convergence of Fourier Series of Functions Which Equal Zero on Some Set”, Wavelet Analysis and Applications, Applied and Numerical Harmonic Analysis, ed. Tao Q. Mang V. Xu Y., Birkhauser Boston, 2007, 13–24
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