Abstract
The convergence is established of the expansion of functions of H αp not satisfying any boundary conditions, in Fourier series with respect to a fundamental system of functions of the Laplace operator in any two-dimensional region with a rectifiable boundary.
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S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [in Russian], Moscow (1969).
V. A. Il'in and Sh. A. Alimov, “Conditions for the Riesz summability of Fourier series with respect to any fundamental set of functions of the Laplace operator, best-possible in the Sobolev, Nikol'skii, Besov, Liouville, and Zygmund-Holder classes,” Dokl. Akad. Nauk SSSR,193, No. 2, 267–269 (1970).
V. A. Il'in, “A result concerning the expandibility of a piecewise-smooth function in a series of eigen-functions for an arbitrary two-dimensional region,” Dokl. Akad. Nauk SSSR,109, No. 3, 442–445 (1956).
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Translated from Matematicheskie Zametki, Vol. 9, No. 6, pp. 609–616, June, 1971.
The author expresses his gratitude to V. A. II'in for suggesting this problem and for his interest in the work.
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Alimov, S.A. Expandibility in eigenfunctions of the Laplacian in a two-dimension region. Mathematical Notes of the Academy of Sciences of the USSR 9, 353–357 (1971). https://doi.org/10.1007/BF01094575
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DOI: https://doi.org/10.1007/BF01094575