Vector-valued Lagrange interpolation and mean convergence of Hermite series

Abstract

Let X be a Banach space and $1\le p<\infty$. We prove interpolation inequalities of Marcinkiewicz-Zygmund type for X-valued polynomials g of degree $\le n$ on $R$, \[c_p (\sum\limits_{i=1}^{n+1} \mu_i \| g(t_i)e^{-t_i^2 /2} \|^p)^{1/p} \le (\int\limits_{\RR}^{} \|g(t)e^{-t^2 /2} \|^p dt)^{1/p} \le d_p (\sum\limits_{i=1}^{n+1} \mu_i \|g(t_i)e^{-t_i^2 /2} \|^p)^{1/p}\;\;,\] where $(t_i)_1^{n+1}$ are the zeros of the Hermite polynomial $H_{n+1}$ and $(\mu_i)_1^{n+1}$ are suitable weights. The validity of the right inequality requires $1<p<4$ and X being a UMD-space. This implies a mean convergence theorem for the Lagrange interpolation polynomials of continuous functions on $R$ taken at the zeros of the Hermite polynomials. In the scalar case, this improves a result of Nevai $[$N$]$. Moreover, we give vector-valued extensions of the mean convergence results of Askey-Wainger $[$AW$]$ in the case of Hermite expansions.