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Uspekhi Mat. Nauk, 1976, Volume 31, Issue 6(192), Pages 28–83 (Mi umn3993)  

This article is cited in 21 scientific papers (total in 22 papers)

Convergence problems of multiple trigonometric series and spectral decompositions. I

Sh. A. Alimov, V. A. Il'in, E. M. Nikishin


Abstract: This article is a survey of the present state of convergence problems of multiple trigonometric series and spectral decompositions corresponding to self-adjoint elliptic differential operators. Particular attention is paid to questions of localization, uniform convergence, convergence in mean, convergence almost everywhere, and also of absolute convergence. In some cases short sketches of proofs of important propositions are given. This applies mostly to Chapter III, which contains a large number of new proofs of various propositions connected with convergence problems of spectral decompositions.
In addition, the article mentions unsolved questions and formulates many new problems

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English version:
Russian Mathematical Surveys, 1976, 31:6, 29–86

Bibliographic databases:

UDC: 517.4+517.5
MSC: 42A20, 42A32, 42A24
Received: 11.02.1975

Citation: Sh. A. Alimov, V. A. Il'in, E. M. Nikishin, “Convergence problems of multiple trigonometric series and spectral decompositions. I”, Uspekhi Mat. Nauk, 31:6(192) (1976), 28–83; Russian Math. Surveys, 31:6 (1976), 29–86

Citation in format AMSBIB
\Bibitem{AliIliNik76}
\by Sh.~A.~Alimov, V.~A.~Il'in, E.~M.~Nikishin
\paper Convergence problems of multiple trigonometric series and spectral decompositions.~I
\jour Uspekhi Mat. Nauk
\yr 1976
\vol 31
\issue 6(192)
\pages 28--83
\mathnet{http://mi.mathnet.ru/umn3993}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=461028}
\zmath{https://zbmath.org/?q=an:0345.42005|0367.42008}
\transl
\jour Russian Math. Surveys
\yr 1976
\vol 31
\issue 6
\pages 29--86
\crossref{https://doi.org/10.1070/RM1976v031n06ABEH001575}


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    This publication is cited in the following articles:
    1. 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
    2. Christopher Meaney, “On almost-everywhere convergent eigenfunction expansions of the Laplace–Beltrami operator”, Math Proc Camb Phil Soc, 92:1 (1982), 129  crossref  mathscinet  zmath  isi
    3. Donald I Cartwright, Paolo M Soardi, “Best conditions for the norm convergence of Fourier series”, Journal of Approximation Theory, 38:4 (1983), 344  crossref
    4. B Dreseler, “On the asymptotics of the Bochner-Riesz kernel on compact Lie groups”, Journal of Functional Analysis, 68:3 (1986), 374  crossref
    5. A. G. Vitushkin, A. A. Gonchar, B. S. Kashin, A. I. Kostrikin, S. M. Nikol'skii, S. P. Novikov, P. L. Ul'yanov, L. D. Faddeev, “Evgenii Mikhailovich Nikishin (obituary)”, Russian Math. Surveys, 42:5 (1987), 153–160  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. Leonardo Colzani, “Riesz means of eigenfunction expansions of elliptic differential operators on compact manifolds”, Seminario Mat e Fis di Milano, 58:1 (1988), 149  crossref  mathscinet  zmath
    7. Leonardo Colzani, Giancarlo Travaglini, “Estimates for Riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds”, Journal of Functional Analysis, 96:1 (1991), 1  crossref
    8. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. D. A. Popov, “Estimates with constants for some classes of oscillatory integrals”, Russian Math. Surveys, 52:1 (1997), 73–145  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    10. E. D. Nursultanov, “Concerning the multiplicators of Fourier series in the trigonometric system”, Math. Notes, 63:2 (1998), 205–214  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. D. A. Popov, “Reconstruction of characteristic functions in two-dimensional Radon tomography”, Russian Math. Surveys, 53:1 (1998), 109–193  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    12. V. A. Yudin, “Multidimensional Versions of Paley's Inequality”, Math. Notes, 70:6 (2001), 860–865  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. V. A. Yudin, “On Positive Values of Spherical Harmonics and Trigonometric Polynomials”, Math. Notes, 75:3 (2004), 447–450  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. Sh. A. Alimov, “Dependence of the Convergence Domain of Spectral Expansions on the Geometry of the Set of Discontinuity of the Function Being Expanded”, Math. Notes, 79:2 (2006), 165–177  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. I. L. Bloshanskii, T. A. Matseevich, “A Weak Generalize Localization of Multiple Fourier Series of Continuous Functions with a Certain Module of Continuity”, Journal of Mathematical Sciences, 155:1 (2008), 31–46  mathnet  crossref  mathscinet  zmath
    16. V. A. Mikhailets, A. A. Murach, “Elliptic boundary-value problem in a two-sided improved scale of spaces”, Ukr Math J, 60:4 (2008), 574  crossref  mathscinet  isi
    17. Ravshan R. Ashurov, “On summability of eigenfunction expansions of piecewise smooth functions”, Applicable Analysis, 87:1 (2008), 29  crossref
    18. Anvarjon Ahmedov, “The principle of general localization on unit sphere”, Journal of Mathematical Analysis and Applications, 356:1 (2009), 310  crossref
    19. A. A. Rakhimov, “On the Localization of the Riesz Means of Multiple Fourier Series of Distributions”, Abstract and Applied Analysis, 2011 (2011), 1  crossref
    20. Alimov Sh.A., “O gladkosti srednikh znachenii funktsii s summiruemym spektralnym razlozheniem”, Differentsialnye uravneniya, 48:4 (2012), 498–498  elib
    21. O. I. Kuznetsova, A. N. Podkorutov, “On the Norms of the Integral Means of Spherical Fourier Sums”, Math. Notes, 96:5 (2014), 690–697  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    22. A. A. Rakhimov, “On the uniform convergence of Fourier series on a closed domain”, Eurasian Math. J., 8:3 (2017), 60–69  mathnet
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