Approximation theory

Daghestan Electronic Mathematical Reports: Issue 4 (2015)

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

UDK: 517.538

Pages: 1 - 14

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.

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