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Russian Mathematical Surveys, 1968, Volume 23, Issue 2, Pages 59–116
DOI: https://doi.org/10.1070/RM1968v023n02ABEH001238
(Mi rm5610)
 

This article is cited in 25 scientific papers (total in 25 papers)

Problems of localization and convergence for Fourier series in fundamental systems of functions of the Laplace operator

V. A. Il'in
References:
Abstract: The paper deals with the problems of localization and convergence of Fourier series with respect to a so-called $fundamental system of the Laplace operator$. (As understood by the author in this paper, a fundamental system of functions includes the eigenfunctions of all boundary-value problems and is characterized by the absence of any kind of boundary conditions).
Chapter 1 of the paper contains a survey of all the important results on the problems of localization and convergence of Fourier series, both for concrete systems of eigenfunctions of the Laplace operator (and, in particular, for the multiple trigonometric system) and for arbitrary fundamental systems of functions of this operator.
In Chapters 2–5 detailed proofs are given for the recent results of the author concerning general fundamental systems of functions of the Laplace operator, including: 1) a comprehensive solution of the localization problem for an arbitrary N-dimensional domain in the Sobolev classes Wα2 (with non-integral α), 2) a comprehensive solution of the localization and convergence problem for an arbitrary odd-dimensional domain in the Hölder classes C(n,α), 3) almost definitive conditions for localization and convergence for an arbitrary even-dimensional domain, 4) a proof of the result that in the class of all N-dimensional domains the smoothness conditions prescribed for the function f(x) to be expanded are best possible even for an arbitrary rearrangement of the terms of the Fourier series.
Received: 26.07.1967
Bibliographic databases:
Document Type: Article
UDC: 517.432+517.512/4
Language: English
Original paper language: Russian
Citation: V. A. Il'in, “Problems of localization and convergence for Fourier series in fundamental systems of functions of the Laplace operator”, Russian Math. Surveys, 23:2 (1968), 59–116
Citation in format AMSBIB
\Bibitem{Ili68}
\by V.~A.~Il'in
\paper Problems of localization and convergence for Fourier series in fundamental systems of functions of the Laplace operator
\jour Russian Math. Surveys
\yr 1968
\vol 23
\issue 2
\pages 59--116
\mathnet{http://mi.mathnet.ru/eng/rm5610}
\crossref{https://doi.org/10.1070/RM1968v023n02ABEH001238}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=223823}
\zmath{https://zbmath.org/?q=an:0189.35702}
Linking options:
  • https://www.mathnet.ru/eng/rm5610
  • https://doi.org/10.1070/RM1968v023n02ABEH001238
  • https://www.mathnet.ru/eng/rm/v23/i2/p61
  • This publication is cited in the following 25 articles:
    1. M. V. Suchkov, V. P. Trifonenkov, “Ob absolyutnoi skhodimosti spektralnykh razlozhenii v dvumernoi zamknutoi oblasti dlya operatora Laplasa s razryvnym koeffitsientom i zadachi Dirikhle”, Mezhdunar. nauch.-issled. zhurn., 2023, no. 3(129), 1–6  mathnet  crossref
    2. K. I. Babenko, “On the mean convergence of multiple Fourier series and the asymptotics of the Dirichlet kernel of spherical means”, Eurasian Math. J., 9:4 (2018), 22–60  mathnet  crossref
    3. E. Liflyand, “Babenko's work on spherical Lebesgue constants”, Eurasian Math. J., 9:4 (2018), 79–81  mathnet  crossref
    4. A Fargana, A A Rakhimov, A A Khan, T B H Hassan, “Equiconvergence in Summation Associated with Elliptic Polynomial”, J. Phys.: Conf. Ser., 949 (2017), 012001  crossref
    5. O. I. Kuznetsova, A. N. Podkorytov, “On strong averages of spherical Fourier sums”, St. Petersburg Math. J., 25:3 (2014), 447–453  mathnet  crossref  mathscinet  zmath  isi  elib
    6. Goldman M.L., “Optimal embedding of Bessel- and Riesz-type potentials”, Dokl. Math., 80:2 (2009), 689–693  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    7. Anvarjon Ahmedov, “The principle of general localization on unit sphere”, Journal of Mathematical Analysis and Applications, 356:1 (2009), 310  crossref
    8. Anthony Carbery, Fernando Soria, Ana Vargas, “Localisation and weighted inequalities for spherical Fourier means”, J Anal Math, 103:1 (2007), 133  crossref  mathscinet  zmath  isi
    9. M. I. Dyachenko, “U-Convergence of Fourier Series with Monotone and with Positive Coefficients”, Math. Notes, 70:3 (2001), 320–328  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. Encyclopaedia of Mathematics, Supplement III, 2001, 234  crossref
    11. Anthony Carbery, Fernando Soria, “Pointwise Fourier inversion and localisation in Rn ”, The Journal of Fourier Analysis and Applications, 3:s1 (1997), 847  crossref  mathscinet  zmath  isi
    12. A CARBERY, F SORIA, “Sets of divergence for the localization problem for Fourier integrals”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 325:12 (1997), 1283  crossref
    13. Yu. N. Subbotin, “The Lebesgue constants of certain m-dimensional interpolation polynomials”, Math. USSR-Sb., 46:4 (1983), 561–570  mathnet  crossref  mathscinet  zmath
    14. R. M. Trigub, “Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus”, Math. USSR-Izv., 17:3 (1981), 567–593  mathnet  crossref  mathscinet  zmath  isi
    15. H.J Mertens, R.J Nessel, “An equivalence theorem concerning multipliers of strong convergence”, Journal of Approximation Theory, 30:4 (1980), 284  crossref
    16. H. J. Mertens, R. J. Nessel, “Über Multiplikatoren starker Konvergenz für Fourier-Entwicklungen in Banach-Räumen”, Math Nachr, 84:1 (1978), 185  crossref  mathscinet  zmath
    17. V. V. Tikhomirov, “On the Riesz means of expansion in eigenfunctions and associated functions of a nonselfadjoint ordinary differntial operator”, Math. USSR-Sb., 31:1 (1977), 29–48  mathnet  crossref  mathscinet  zmath  isi
    18. Sh. A. Alimov, V. A. Il'in, E. M. Nikishin, “Problems of convergence of multiple trigonometric series and spectral decompositions. II”, Russian Math. Surveys, 32:1 (1977), 115–139  mathnet  crossref  mathscinet  zmath
    19. Sh. A. Alimov, V. A. Il'in, E. M. Nikishin, “Convergence problems of multiple trigonometric series and spectral decompositions. I”, Russian Math. Surveys, 31:6 (1976), 29–86  mathnet  crossref  mathscinet  zmath
    20. H.S Shapiro, “Lebesgue constants for spherical partial sums”, Journal of Approximation Theory, 13:1 (1975), 40  crossref
    Citing articles in Google Scholar: Russian citations, English citations
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