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This article is cited in 5 scientific papers (total in 5 papers)
Sobolev orthogonal polynomials, associated with the Chebyshev polynomials of the first kind
I. I. Sharapudinovab, M. G. Magomed-Kasumovab, S. R. Magomedova a Daghestan Scientific Centre of RAS
b Vladikavkaz Scientific Centre of the RAS
Abstract:
Using Chebyshev polynomials $T_n(x)=\cos(n\arccos x) (n=0,1,\ldots)$, for any natural $r$ we build 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.
Received: 07.10.2015 Revised: 18.11.2015 Accepted: 19.11.2015
Citation:
I. I. Sharapudinov, M. G. Magomed-Kasumov, S. R. Magomedov, “Sobolev orthogonal polynomials, associated with the Chebyshev polynomials of the first kind”, Daghestan Electronic Mathematical Reports, 2015, no. 4, 1–14
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https://www.mathnet.ru/eng/demr15 https://www.mathnet.ru/eng/demr/y2015/i4/p1
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