Jacobi–Sobolev orthogonal polynomials: Asymptotics and a Cohen type inequality

Abstract

Let $d{\mu }_{\alpha ,\beta }\left(x\right)={(1-x)}^{\alpha }{(1+x)}^{\beta }dx$,$\alpha ,\beta >-1$, be the Jacobi measure supported on the interval $[-1,1]$. Let us introduce the Sobolev inner product

$${〈f,g〉}_S=\sum _{j=0}^N{\lambda }_j{\int }_{-1}^1{f}^{\left(j\right)}\left(x\right)g^{\left(j\right)}\left(x\right)d{\mu }_{\alpha ,\beta }\left(x\right)\text{,}$$

where ${\lambda }_j\ge 0$ for $0\le j\le N-1$ and ${\lambda }_N>0$. In this paper we obtain some asymptotic results for the sequence of orthogonal polynomials with respect to the above Sobolev inner product. Furthermore, we prove a Cohen type inequality for Fourier expansions in terms of such polynomials.