### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 4 (2015)

# Sobolev orthogonal polynomials, associated with the Chebyshev polynomials of the first kind

Sharapudinov Idris Idrisovich, Magomed-Kasumov Magomedrasul Grozbekovich, Magomedov Salikh Rasulovich

### UDK: 517.538

### Pages: 1 - 14

### DOI: 10.31029/demr.4.1

Using Chebyshev polynomials $T_n(x)=\cos(n\arccos x) (n=0,1,\ldots)$, for any natural $r$ a new system of polynomials $\left\{T_{r,k}(x)\right\}_{k=0}^\infty$, orthonormal with respect to the Sobolev type inner product of the following form $<f,g>=\sum_{\nu=0}^{r-1}f^{(\nu)}(-1)g^{(\nu)}(-1)+\int_{-1}^{1} f^{(r)}(t)g^{(r)}(t)\kappa(t) dt$, where $\kappa(t)=\frac2\pi(1-t^2)^{-\frac12}$. The convergence of the Fourier series by the system $\left\{T_{r,k}(x)\right\}_{k=0}^\infty$ is investigated. We consider the important special cases of systems of this type. For these instances we obtain explicit representations,
that can be used in the study of asymptotic properties of functions $T_{r,k}(x)$ when $k\to\infty$ and study of the approximative properties of Fourier sums by
the system $\left\{T_{r,k}(x)\right\}_{k = 0}^\infty$.

**Keywords: **
orthogonal polynomials, Sobolev orthogonal polynomials, Chebyshev polynomials of the first kind.