caa a22 4500 37887859X CHVBK 20180305123426.0 cr unu---uuuuu 161128e200309 xx s 000 0 eng 10.1515/GMJ.2003.401 doi (NATIONALLICENCE)gruyter-10.1515/GMJ.2003.401 On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems [Elektronische Daten] [M. S. Agranovich, B. A. Amosov] We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 - 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 - 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides. © Heldermann Verlag Elliptic boundary value problems nationallicence Fourier expansions in eigenfunctions nationallicence Sobolev spaces nationallicence unconditional convergence nationallicence Agranovich M. S. Moscow Institute of Electronics and Mathematics, Moscow 109028, Russia. E-mail: msa.funcan@mtu-net.ru aut Amosov B. A. Moscow Institute of Electronics and Mathematics, Moscow 109028, Russia. E-mail: amosov@orc.ru aut Georgian Mathematical Journal Walter de Gruyter GmbH & Co. KG 10/3(2003-09), 401-410 1072-947X 10:3<401 2003 10 GMJ https://doi.org/10.1515/GMJ.2003.401 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.1515/GMJ.2003.401 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Agranovich M. S. Moscow Institute of Electronics and Mathematics, Moscow 109028, Russia. E-mail: msa.funcan@mtu-net.ru aut NATIONALLICENCE 700 1- Amosov B. A. Moscow Institute of Electronics and Mathematics, Moscow 109028, Russia. E-mail: amosov@orc.ru aut NATIONALLICENCE 773 0- Georgian Mathematical Journal Walter de Gruyter GmbH & Co. KG 10/3(2003-09), 401-410 1072-947X 10:3<401 2003 10 GMJ CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter