Approximation theory

Daghestan Electronic Mathematical Reports: Issue 12 (2019)

On uniform convergence of Fourier-Sobolev series

UDK: 517.538

Pages: 55 - 61

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.

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