Asymptotic properties of polynomials, orthogonal in Sobolev sence and associated with the Jacobi polynomials

I. I. Sharapudinov

Daghestan Electronic Mathematical Reports, Tome 6 (2016), pp. 1-24

Résumé

We consider polynomials $p_{r,n}^{\alpha,\beta}(x)$ $(n=0,1,\ldots)$, generated by classical Jacobi polynomials $p_{n}^{\alpha,\beta}(x)$ and forming orthonormal system with respect to Sobolev-type inner product \begin{equation*} ,g>=\sum_{\nu=0}^{r-1}f^{(\nu)}(-1)g^{(\nu)}(-1)+\int_{-1}^{1}f^{(r)}(t)g^{(r)}(t)\rho(t) dt, \end{equation*} where $\rho(x)=(1-x)^\alpha(1+x)^\beta$ – Jacobi weight function. The explicit \linebreak representations for polynomials $p_{r,n}^{\alpha,\beta}(x)$ are obtained and using these ones the asymptotic properties of polynomials $p_{r,n}^{\alpha,\beta}(x)$ are investigated.

Détail

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