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 Daghestan Electronic Mathematical Reports, 2019, Issue 12, Pages 55–61 (Mi demr77)

On uniform convergence of Fourier-Sobolev series

T. N. Shakh-Emirov

Department of Mathematics and Informatics, DFRC, Makhachkala

Abstract: Let $\{\varphi_{k}\}_{k=0}^\infty$ be a system of functions defined on $[a, b]$ and orthonormal in $L ^ 2_ \rho = L ^ 2_\rho ( a, b)$ with respect to the usual inner product. For a given positive integer $r$, by $\{\varphi_{r,k}\}_{k=0}^\infty$ we denote the system of functions orthonormal with respect to the Sobolev-type inner product and generated by the system $\{\varphi_{k}\}_{k=0}^\infty$. In this paper, we study the question of the uniform convergence of the Fourier series by the system of functions $\{\varphi_{r,k}\}_{k=0}^\infty$ to the functions $f\in W^r_{L^p_\rho}$ in the case when the original system $\{\varphi_{k}\}_{k=0}^\infty$ forms a basis in the space $L^p_\rho=L^p_\rho(a,b)$ ($1\le p$, $p\neq2$).

Keywords: Fourier series; Sobolev-type inner product; Sobolev space; Sobolev-orthonormal functions

DOI: https://doi.org/10.31029/demr.10.7

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UDC: 517.538
Revised: 26.09.2019
Accepted:27.09.2019

Citation: T. N. Shakh-Emirov, “On uniform convergence of Fourier-Sobolev series”, Daghestan Electronic Mathematical Reports, 2019, no. 12, 55–61

Citation in format AMSBIB
\Bibitem{Sha19} \by T.~N.~Shakh-Emirov \paper On uniform convergence of Fourier-Sobolev series \jour Daghestan Electronic Mathematical Reports \yr 2019 \issue 12 \pages 55--61 \mathnet{http://mi.mathnet.ru/demr77} \crossref{https://doi.org/10.31029/demr.10.7}