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Uspekhi Mat. Nauk, 1968, Volume 23, Issue 2(140), Pages 61–120 (Mi umn5610)  

This article is cited in 21 scientific papers (total in 21 papers)

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

V. A. Il'in

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 $W_2^\alpha$ (with non-integral $\alpha$), 2) a comprehensive solution of the localization and convergence problem for an arbitrary odd-dimensional domain in the Hölder classes $C^{(n,\alpha)}$, 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.

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English version:
Russian Mathematical Surveys, 1968, 23:2, 59–116

Bibliographic databases:

UDC: 517.432+517.512/4
MSC: 42A63, 42A20, 26A16, 46E35, 40A30
Received: 26.07.1967

Citation: V. A. Il'in, “Problems of localization and convergence for Fourier series in fundamental systems of functions of the Laplace operator”, Uspekhi Mat. Nauk, 23:2(140) (1968), 61–120; Russian Math. Surveys, 23:2 (1968), 59–116

Citation in format AMSBIB
\by V.~A.~Il'in
\paper Problems of localization and convergence for Fourier series in fundamental systems of functions of the Laplace operator
\jour Uspekhi Mat. Nauk
\yr 1968
\vol 23
\issue 2(140)
\pages 61--120
\jour Russian Math. Surveys
\yr 1968
\vol 23
\issue 2
\pages 59--116

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    This publication is cited in the following articles:
    1. A. A. Arsen'ev, “Asymptotics of the wave function of the quasistationary state”, Theoret. and Math. Phys., 1:1 (1969), 94–107  mathnet  crossref  mathscinet
    2. M. M. Gekhtman, “Spectrum of some nonclassical self-adjoint extensions of the Laplace operator”, Funct. Anal. Appl., 4:4 (1970), 325–326  mathnet  crossref  mathscinet  zmath
    3. B. I. Golubov, “On the convergence of Riesz spherical means of multiple Fourier series and integrals of functions of bounded generalized variation”, Math. USSR-Sb., 18:4 (1972), 635–658  mathnet  crossref  mathscinet  zmath
    4. L. V. Zhizhiashvili, “Some problems in the theory of simple and multiple trigonometric and orthogonal series”, Russian Math. Surveys, 28:2 (1973), 65–127  mathnet  crossref  mathscinet  zmath
    5. H.S Shapiro, “Lebesgue constants for spherical partial sums”, Journal of Approximation Theory, 13:1 (1975), 40  crossref
    6. 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
    7. 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
    8. 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
    9. 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
    10. 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
    11. H.J Mertens, R.J Nessel, “An equivalence theorem concerning multipliers of strong convergence”, Journal of Approximation Theory, 30:4 (1980), 284  crossref
    12. Yu. N. Subbotin, “The Lebesgue constants of certain $m$-dimensional interpolation polynomials”, Math. USSR-Sb., 46:4 (1983), 561–570  mathnet  crossref  mathscinet  zmath
    13. 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
    14. 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
    15. 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
    16. 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
    17. 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
    18. Anvarjon Ahmedov, “The principle of general localization on unit sphere”, Journal of Mathematical Analysis and Applications, 356:1 (2009), 310  crossref
    19. 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
    20. 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
    21. E. Liflyand, “Babenko's work on spherical Lebesgue constants”, Eurasian Math. J., 9:4 (2018), 79–81  mathnet  crossref
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