An asymptotic development of the Poisson integral for Laguerre polynomial expansions

Abstract

The purpose of this paper is the study of the rate of convergence of Poisson integrals for Laguerre expansions. The convergence of partial sums of Fourier series of functions in $L^p$ spaces was studied, for several classes of orthogonal polynomials. In the Laguerre case Askey and Waigner proved convergence for functions $f\in L^p\left(0,+\infty \right)$ with $4/3<p<4$. In this paper we deal with the Poisson integral $A_{r}f$ $\left(0<r<1\right)$ which arises by applying Abel’s summation method to the Laguerre expansion of the function $f$. About 50 years ago, Muckenhoupt intensively studied the Poisson integral for the Laguerre and Hermite polynomials. Among other things he proved pointwise convergence, the convergence by norm, and that the Poisson integral is a contraction mapping in $L^p\left(0,\infty \right)$. Toczek and Wachnicki gave a Voronovskaja-type theorem by calculating the limit ${\left(1-r\right)}^{-1}\left(\left(A_{r}f\right)\left(x\right)-f\left(x\right)\right)$ as $r\to 1^{-}$, provided that $f^{\prime \prime }\left(x\right)$ exists. We generalize this formula by deriving a complete asymptotic development. All its coefficients are explicitly given in a concise form. As an application we apply extrapolation methods in order to improve the rate of convergence of $\left(A_{r}f\right)\left(x\right)$ as $r\to 1^{-}$.