Approximation theory

Daghestan Electronic Mathematical Reports: Issue 6 (2016)


Systems of functions orthogonal with respect to scalar products of Sobolev type with discrete masses generated by classical orthogonal systems

UDK: 517.538

Pages: 31 - 60


For a given system of functions $\left\{\varphi_k(x)\right\}_{k=0}^\infty$ orthonormal on $(a, b)$ with weight $\rho(x)$ and a natural $r$ associated with it the new system of functions $\left\{\varphi_{r,k}(x)\right\}_{k=0}^\infty$ is constructed, orthonormal with respect to the Sobolev type inner product of the following form 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 $\langle f,g\rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b} f^{(r)}(t)g^{(r)}(t)\rho(t) dt.$ The convergence of the Fourier series by the system $\left\{\varphi_{r,k}(x)\right\}_{k=0}^\infty$ is investigated. We consider the important special cases of systems type $\left\{\varphi_{r,k}(x)\right\}_{k=0}^\infty$, for which instances we obtained explicit representations, that can be used in the study of asymptotic properties of functions $\varphi_{r,k}(x)$ when $k\to\infty$ and study of the approximative properties of Fourier sums by the system $\left\{\varphi_{r,k}(x)\right\}_{k = 0}^\infty$. 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.




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