Hermite expansions of C0-groups and cosine functions

Abstract

In this paper we introduce vector-valued Hermite expansions to approximate oneparameter operator families such as C 0 -groups and cosine functions. In both cases we estimate the rate of convergence of these Hermite expansions to the related family and compare to other already known approximations. Finally, we illustrate our results with particular examples of C 0 -groups and cosine functions and their Hermite expansions.

Quadrature Rules for L 1-Weighted Norms of Orthogonal Polynomials

“…The family {h n } n≥0 is a total set in L p (R), with 1 ≤ p < ∞, and the optimal estimate of h n 1 has a great importance on the study of vector-valued Hermite expansions, see more details in [2]. By standard techniques, the known bound…”

“…These functions q n,ω are fundamental in classical orthogonal expansions ([11, Chapter 4] and [14, Chapter IX]). Recently the authors have treated them to introduce Laguerre expansions for C 0 -semigroups in [1] and Hermite expansions for C 0 -groups and cosine function in [2]. In fact, to get sharp estimations of q n,ω 1 is the motivating starting-point of this paper: sharp estimations allow to assure convergence of vector-valued orthogonal expansions, see more details in [1,2].…”