$C_0$-semigroups and resolvent operators approximated by Laguerre expansions

TL;DR

This paper develops Laguerre expansion methods to approximate $C_0$-semigroups and resolvent operators in Banach spaces, extending scalar results and providing convergence analysis with applications to various semigroup examples.

Contribution

It introduces Laguerre expansions for vector-valued functions and applies them to approximate $C_0$-semigroups and resolvent operators, with convergence results and specific examples.

Findings

Laguerre series converge in Lebesgue spaces for the studied functions.

Approximation methods are improved for shift, convolution, and holomorphic semigroups.

Theoretical convergence results are established for the Laguerre expansion approximations.

Abstract

In this paper we introduce Laguerre expansions to approximate vector-valued functions expanding on the well-known scalar theorem. We apply this result to approximate $C_0$-semi\-groups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions, its Laplace transforms and the convergence of Laguerre series in Lebesgue spaces. The concluding section of this paper is devote to consider some examples of $C_0$-semigroups: shift, convolution and holomorphic semigroups where some of these results are improved.