### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 12 (2019)

# On uniform convergence of Fourier-Sobolev series

### UDK: 517.538

### Pages: 55 - 61

### DOI: 10.31029/demr.12.5

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.