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

This article is cited in 3 scientific papers (total in 3 papers)

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

Full text: PDF file (287 kB)
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English version:
Sbornik: Mathematics, 2000, 191:11, 1587–1606

Bibliographic databases:

UDC: 517.51
MSC: 42B08, 42B10
Received: 10.01.2000

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
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\pages 3--20
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\pages 1587--1606
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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  crossref  isi
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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