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This article is cited in 7 scientific papers (total in 7 papers)
Systems of functions orthogonal with respect to scalar products of Sobolev type with discrete masses generated by classical orthogonal systems
I. I. Sharapudinovab, Z. D. Gadzhievaab, R. M. Gadzhimirzaeva a Daghestan Scientific Centre of Russian Academy of Sciences
b Daghestan State Pedagogical University
Abstract:
For some natural number $r$ and a given system of functions $\left\{\varphi_k(x)\right\}_{k=0}^\infty$, orthonormal on $(a, b)$ with weight $\rho(x)$,
we construct the new system of functions $\left\{\varphi_{r,k}(x)\right\}_{k=0}^\infty$, orthonormal with respect to the Sobolev type inner product of the following form
\begin{equation*}
\langle f,g\rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b} f^{(r)}(t)g^{(r)}(t)\rho(t) dt.
\end{equation*}
The convergence of the Fourier series by the system $\left\{\varphi_{r,k}(x)\right\}_{k=0}^\infty$ is investigated.
Moreover, we consider some important special cases of systems of such type and obtain explicit representations for them, which can be used in the study of asymptotic properties of functions $\varphi_{r,k}(x)$ when $k\to\infty$ and the approximative properties of Fourier sums by the system $\left\{\varphi_{r,k}(x)\right\}_{k = 0}^\infty$.
Keywords:
orthogonal polynomials, Sobolev orthogonal polynomials, Haar system, Jacobi polynomials, Сhebyshev polynomials of the first kind, Laguerre polynomials, Hermite polynomials.
Received: 29.07.2016 Revised: 07.09.2016 Accepted: 08.09.2016
Citation:
I. I. Sharapudinov, Z. D. Gadzhieva, R. M. Gadzhimirzaev, “Systems of functions orthogonal with respect to scalar products of Sobolev type with discrete masses generated by classical orthogonal systems”, Daghestan Electronic Mathematical Reports, 2016, no. 6, 31–60
Linking options:
https://www.mathnet.ru/eng/demr28 https://www.mathnet.ru/eng/demr/y2016/i6/p31
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