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Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2015, Volume 15, Issue 4, Pages 405–418 (Mi isu608)  

This article is cited in 1 scientific paper (total in 1 paper)

Mathematics

Estimates of speed of convergence and equiconvergence of spectral decomposition of ordinary differential operators

I. S. Lomov

Moscow State University, Lenin mountains, 119992, Moscow, Russia

Abstract: The present review contains results of V. A. Il'in and his pupils concerning an assessment of speed of convergence and equiconvergence with a trigonometrical series of Fourier of spectral decomposition of functions on root functions of linear ordinary differential operators both self-conjugate, and not self-conjugate, set on a final piece of a numerical straight line. The first theorem of V. A. Ilyin of equiconvergence of spectral decomposition for the differential operator of any order is provided. Theorems of the speed of equiconvergence of spectral decomposition at first for any self-conjugate expansions of the one-dimensional operator Schrodinger are formulated. Thus the potential of the operator can have any features on interval border. This allows us to receive new results even for all classical orthogonal polynomials. Further results for not self-conjugate operators are formulated. The review for the so-called loaded differential operators comes to the end with the theorem of equiconvergence speed. Estimates of speed of equiconvergence of decomposition are received both on any internal compact of an interval, and on the whole interval. Dependence of an assessment of speed of equiconvergence of decomposition on any compact of the main interval from distance of this compact to interval border is established.

Key words: ordinary differential operator, eigenvalues, spectral decomposition, convergence speed, formula of average value.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-12472-офи-м
This work was supported by the Russian Foundation for Basic Research (projects no. 13-01-12472).


DOI: https://doi.org/10.18500/1816-9791-2015-15-4-405-418

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UDC: 517.927.25

Citation: I. S. Lomov, “Estimates of speed of convergence and equiconvergence of spectral decomposition of ordinary differential operators”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 15:4 (2015), 405–418

Citation in format AMSBIB
\Bibitem{Lom15}
\by I.~S.~Lomov
\paper Estimates of speed of convergence and equiconvergence of spectral decomposition of ordinary differential operators
\jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.
\yr 2015
\vol 15
\issue 4
\pages 405--418
\mathnet{http://mi.mathnet.ru/isu608}
\crossref{https://doi.org/10.18500/1816-9791-2015-15-4-405-418}
\elib{http://elibrary.ru/item.asp?id=25360656}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. S. Lomov, “Spektralnyi metod Ilina ustanovleniya svoistv bazisnosti i ravnomernoi skhodimosti biortogonalnykh razlozhenii na konechnom intervale”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 19:1 (2019), 34–58  mathnet  crossref
  • Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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