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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Volume 261, Pages 249–257 (Mi tm753)  

This article is cited in 2 scientific papers (total in 2 papers)

Equiconvergence of the Trigonometric Fourier Series and the Expansion in Eigenfunctions of the Sturm–Liouville Operator with a Distribution Potential

I. V. Sadovnichaya

M. V. Lomonosov Moscow State University
Full-text PDF (190 kB) Citations (2)
References:
Abstract: We consider the Sturm–Liouville operator $L=-d^2/dx^2+q(x)$ with the Dirichlet boundary conditions in the space $L_2[0,\pi]$ under the assumption that the potential $q(x)$ belongs to $W_2^{-1}[0,\pi]$. We study the problem of uniform equiconvergence on the interval $[0,\pi]$ of the expansion of a function $f(x)$ in the system of eigenfunctions and associated functions of the operator $L$ and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function $f(x)$ of class $L_1$. We also consider the case of potentials belonging to the scale of Sobolev spaces $W_2^{-\theta}[0,\pi]$ with $\frac12<\theta\le1$. We show that if the antiderivative $u(x)$ of the potential belongs to some space $W_2^\theta[0,\pi]$ with $0<\theta<\frac12$, then, for any function in the space $L_2[0,\pi]$, the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing $u(x)$. We also give an explicit estimate for the rate of equiconvergence.
Received in March 2007
English version:
Proceedings of the Steklov Institute of Mathematics, 2008, Volume 261, Pages 243–252
DOI: https://doi.org/10.1134/S0081543808020193
Bibliographic databases:
UDC: 517.984+517.518.45
Language: Russian
Citation: I. V. Sadovnichaya, “Equiconvergence of the Trigonometric Fourier Series and the Expansion in Eigenfunctions of the Sturm–Liouville Operator with a Distribution Potential”, Differential equations and dynamical systems, Collected papers, Trudy Mat. Inst. Steklova, 261, MAIK Nauka/Interperiodica, Moscow, 2008, 249–257; Proc. Steklov Inst. Math., 261 (2008), 243–252
Citation in format AMSBIB
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\by I.~V.~Sadovnichaya
\paper Equiconvergence of the Trigonometric Fourier Series and the Expansion in Eigenfunctions of the Sturm--Liouville Operator with a~Distribution Potential
\inbook Differential equations and dynamical systems
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2008
\vol 261
\pages 249--257
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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