C0C0-semigroups and resolvent operators approximated by Laguerre expansions

In this paper we introduce Laguerre expansions to approximate vector-valued functions. We apply this result to approximate C0C0-semigroups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions in order to estimate the rate of convergence of these expansions. Finally, we illustrate the main results of this paper with some examples: shift, convolution and holomorphic semigroups, where the rate of convergence is improved.