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 Mat. Sb., 2000, Volume 191, Number 11, Pages 3–20 (Mi msb496)

Convergence of Fourier double series and Fourier integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates

O. S. Dragoshanskii

M. V. Lomonosov Moscow State University

Abstract: Let $f(\xi,\eta)$ be a function vanishing for $\xi^2+\eta^2>r^2$, where $r$ is sufficiently small, and with Fourier series (of the function considered in the square $(-\pi,\pi]^2$) or Fourier integral (of the function considered in the plane $\mathbb R^2$) convergent uniformly or almost everywhere over rectangles. It is shown that a rotation of the system of coordinates through $\pi/4$
$$\begin{cases} \xi=(x-y)/\sqrt 2 , \eta=(y+x)/\sqrt 2 \end{cases}$$
can “damage” the convergence of the Fourier series or the Fourier integral of the resulting function.

DOI: https://doi.org/10.4213/sm496

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English version:
Sbornik: Mathematics, 2000, 191:11, 1587–1606

Bibliographic databases:

UDC: 517.51
MSC: 42B08, 42B10

Citation: O. S. Dragoshanskii, “Convergence of Fourier double series and Fourier integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates”, Mat. Sb., 191:11 (2000), 3–20; Sb. Math., 191:11 (2000), 1587–1606

Citation in format AMSBIB
\Bibitem{Dra00} \by O.~S.~Dragoshanskii \paper Convergence of Fourier double series and Fourier integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates \jour Mat. Sb. \yr 2000 \vol 191 \issue 11 \pages 3--20 \mathnet{http://mi.mathnet.ru/msb496} \crossref{https://doi.org/10.4213/sm496} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1827510} \zmath{https://zbmath.org/?q=an:1018.42006} \transl \jour Sb. Math. \yr 2000 \vol 191 \issue 11 \pages 1587--1606 \crossref{https://doi.org/10.1070/sm2000v191n11ABEH000496} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000168023700001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0034340558} 

• http://mi.mathnet.ru/eng/msb496
• https://doi.org/10.4213/sm496
• http://mi.mathnet.ru/eng/msb/v191/i11/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bloshanskii I.L., “Linear transformations of R-N and problems of convergence of multiple Fourier integral”, Wavelet Analysis and Active Media Technology Vols 1-3, 2005, 1081–1091
2. 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, 2007, 13–24
3. G. G. Oniani, K. A. Chubinidze, “Rotation of coordinate system and differentiation of integrals with respect to translation-invariant bases”, Sb. Math., 208:4 (2017), 510–530
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