### 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

### DOI: 10.31029/demr.6.3

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.