On uniform convergence of Fourier-Sobolev series

T. N. Shakh-Emirov

Daghestan Electronic Mathematical Reports, Tome 12 (2019), pp. 55-61.

Résumé

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

Détail

Mots-clés : Fourier series; Sobolev-type inner product; Sobolev space; Sobolev-orthonormal functions.

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