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Mat. Sb., 2018, Volume 209, Number 9, Pages 142–170 (Mi msb8910)  

Sobolev orthogonal polynomials generated by Jacobi and Legendre polynomials, and special series with the sticking property for their partial sums

I. I. Sharapudinovab

a Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala
b Vladikavkaz Scientific Centre of the Russian Academy of Sciences

Abstract: The paper is concerned with polynomials $p^{\alpha,\beta}_{r,k}(x)$, $k=0,1,…$, orthonormal with respect to the Sobolev-type inner product
$$ \langle f,g\rangle =\sum_{\nu=0}^{r-1}f^{(\nu)}(-1)g^{(\nu)}(-1)+\int_{-1}^{1}f^{(r)}(t)g^{(r)}(t)(1-t)^\alpha(1+t)^\beta  dt , $$
where $r$ is an arbitrary natural number. Fourier series in the polynomials $p_{r,k}(x)=p^{0,0}_{r,k}(x)$ and some generalizations of them are introduced. Partial sums of such series are shown to retain certain important properties of the partial sums of Fourier series in the polynomials $p_{r,k}(x)$, in particular, the property of $r$-fold coincidence (sticking) of the original function $f(x)$ and the partial sums of the Fourier series in the polynomials $p_{r,k}(x)$ at the points $-1$ and $1$. The main emphasis is put on problems of approximating smooth and analytic functions by partial sums of such generalized series, which are special series in ultraspherical Jacobi polynomials, whose partial sums have the sticking property at the points $-1$ and $1$.
Bibliography: 31 titles.

Keywords: Fourier series in Sobolev orthogonal polynomials; Legendre and Jacobi polynomials; special (sticking) series is ultraspherical polynomials; approximative properties.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00486-а
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 16-01-00486-a).


DOI: https://doi.org/10.4213/sm8910

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English version:
Sbornik: Mathematics, 2018, 209:9, 1390–1417

Bibliographic databases:

UDC: 517.538
MSC: 33C45, 42C10
Received: 10.01.2017 and 22.05.2017

Citation: I. I. Sharapudinov, “Sobolev orthogonal polynomials generated by Jacobi and Legendre polynomials, and special series with the sticking property for their partial sums”, Mat. Sb., 209:9 (2018), 142–170; Sb. Math., 209:9 (2018), 1390–1417

Citation in format AMSBIB
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