Magomed-Kasumov, M. G. Sobolev orthogonal systems with two discrete points and Fourier series. (English. Russian original) Zbl 1484.42027 Russ. Math. 65, No. 12, 47-55 (2021); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2021, No. 12, 56-66 (2021). Denote by \(W_{L^2}^{1}=W_{L^2}^{1}[a, b]\) a Sobolev space consisting of absolutely continuous on \([a,b]\) functions \(f\) such that \(f^\prime\in L^2[a, b]\). Let space \(W_{L^2}^1\) be endowed with inner product \[ \langle f, g\rangle_{S} = f(a)g(a) + f(b)g(b) +\int_{a}^{b} f^\prime(t) g^\prime(t) dt. \]Let \(\Phi=\{\varphi_k\}_{k=0}^{\infty}\) be a system of functions from \(L^2([a,b])\) such that \[ \int_{a}^{b}\varphi_0(t)dt\not =0, \qquad \int_{a}^{b}\varphi_k(t)=0, \quad k\geq 1. \]Introduce a new system of functions \(\Phi_1=\{\varphi_{1,k}\}\) defined by: \begin{align*} \varphi_{1,0}(x)&=\frac{1}{\sqrt{2}}, \\ \varphi_{1,1}(x)&=\frac{1}{\sqrt{1+\frac{1}{2}J_0^2}}\left(-\frac{1}{2}J_0+\int_{a}^{x}\varphi_0(t)dt\right),\quad J_0=\int_{a}^{b}\varphi_0(t)dt, \\ \varphi_{1,k}(x)&=\int_{a}^{x}\varphi_{k-1}(t)dt,\quad k\geq 2. \end{align*}In the presented paper it is proved that if the system \(\Phi\) is a complete ortonormal system in \(L^2\) then \(\Phi_1\) is a complete orthonormal system in \(W_{L^2}^{1}\).It is also established that if the system \(\Phi_1\) is a complete orthonormal system in \(W_{L^2}^{1}\) then for any function \(f\in W_{L^2}^{1}\) its Fourier series with respect to the system \(\Phi_1\) uniformly converges to the function itself. Reviewer: Rostom Getsadze (Uppsala) Cited in 1 Document MSC: 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42A20 Convergence and absolute convergence of Fourier and trigonometric series 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:discrete-continuous inner product; Sobolev inner product; Faber-Schauder system; Jacobi polynomials with negative parameters; Fourier series; uniform convergence; coincidence at ends of segment; completeness of Sobolev systems PDFBibTeX XMLCite \textit{M. G. Magomed-Kasumov}, Russ. Math. 65, No. 12, 47--55 (2021; Zbl 1484.42027); translation from Izv. Vyssh. Uchebn. 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