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 Izv. Saratov Univ. Math. Mech. Inform., 2019, Volume 19, Issue 1, Pages 34–58 (Mi isu788)

Scientific Part
Mathematics

The Il'in spectral method for determination of the properties of the basis property and the uniform convergence of biorthogonal expansions on a finite interval

I. S. Lomov

Lomonosov Moscow State University, Leninskie Gory, 119991 Moscow, Russia

Abstract: The paper discusses the basics of the spectral method of V. A. Il'in on an example of a simple second order differential operator on a segment of the number line. The first theorem of Il'in on the unconditional basis property is stated. Its detailed proof is given. A chain of generalizations of this theorem is traced. A recently established a theorem on the unconditional basis property for the differential operators with general integral boundary conditions is formulated. The substantiation of the statements about the uniform convergence of biorthogonal expansions of functions using the Il'in method is presented. The main theorems, including, the recently established theorem for operators with integral boundary conditions are formulated.

Key words: differential operator, eigenfunctions and associated functions, spectrum, unconditional basis, uniform convergence of biorthogonal series.

DOI: https://doi.org/10.18500/1816-9791-2019-19-1-34-58

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Bibliographic databases:

UDC: 517.927.25
Accepted:15.06.2018

Citation: I. S. Lomov, “The Il'in spectral method for determination of the properties of the basis property and the uniform convergence of biorthogonal expansions on a finite interval”, Izv. Saratov Univ. Math. Mech. Inform., 19:1 (2019), 34–58

Citation in format AMSBIB
\Bibitem{Lom19} \by I.~S.~Lomov \paper The Il'in spectral method for determination of the properties of the basis property and the uniform convergence of biorthogonal expansions on a finite interval \jour Izv. Saratov Univ. Math. Mech. Inform. \yr 2019 \vol 19 \issue 1 \pages 34--58 \mathnet{http://mi.mathnet.ru/isu788} \crossref{https://doi.org/10.18500/1816-9791-2019-19-1-34-58} \elib{https://elibrary.ru/item.asp?id=39524581}