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

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

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
\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 Uspekhi Mat. Nauk \yr 1968 \vol 23 \issue 2(140) \pages 61--120 \mathnet{http://mi.mathnet.ru/umn5610} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=223823} \zmath{https://zbmath.org/?q=an:0189.35702} \transl \jour Russian Math. Surveys \yr 1968 \vol 23 \issue 2 \pages 59--116 \crossref{https://doi.org/10.1070/RM1968v023n02ABEH001238} 

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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. A. A. Arsen'ev, “Asymptotics of the wave function of the quasistationary state”, Theoret. and Math. Phys., 1:1 (1969), 94–107
2. M. M. Gekhtman, “Spectrum of some nonclassical self-adjoint extensions of the Laplace operator”, Funct. Anal. Appl., 4:4 (1970), 325–326
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
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
5. H.S Shapiro, “Lebesgue constants for spherical partial sums”, Journal of Approximation Theory, 13:1 (1975), 40
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
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
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
9. H. J. Mertens, R. J. Nessel, “Über Multiplikatoren starker Konvergenz für Fourier-Entwicklungen in Banach-Räumen”, Math Nachr, 84:1 (1978), 185
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
11. H.J Mertens, R.J Nessel, “An equivalence theorem concerning multipliers of strong convergence”, Journal of Approximation Theory, 30:4 (1980), 284
12. Yu. N. Subbotin, “The Lebesgue constants of certain $m$-dimensional interpolation polynomials”, Math. USSR-Sb., 46:4 (1983), 561–570
13. Anthony Carbery, Fernando Soria, “Pointwise Fourier inversion and localisation in Rn ”, The Journal of Fourier Analysis and Applications, 3:s1 (1997), 847
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
15. M. I. Dyachenko, “$U$-Convergence of Fourier Series with Monotone and with Positive Coefficients”, Math. Notes, 70:3 (2001), 320–328
16. Anthony Carbery, Fernando Soria, Ana Vargas, “Localisation and weighted inequalities for spherical Fourier means”, J Anal Math, 103:1 (2007), 133
17. Goldman M.L., “Optimal embedding of Bessel- and Riesz-type potentials”, Dokl. Math., 80:2 (2009), 689–693
18. Anvarjon Ahmedov, “The principle of general localization on unit sphere”, Journal of Mathematical Analysis and Applications, 356:1 (2009), 310
19. O. I. Kuznetsova, A. N. Podkorytov, “On strong averages of spherical Fourier sums”, St. Petersburg Math. J., 25:3 (2014), 447–453
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
21. E. Liflyand, “Babenko's work on spherical Lebesgue constants”, Eurasian Math. J., 9:4 (2018), 79–81
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