\(C_0\)-semigroups and resolvent operators approximated by Laguerre expansions (Q346215)

The authors use Laguerre expansions to approximate vector-valued functions in abstract Banach spaces. The proof is similar to a theorem to the scalar case shown in [Ann. Math. (2) 28, 593--619 (1927; JFM 53.0270.01)] by \textit{J. V. Uspensky}. They also represent \(C_{0}\)-semigroups and resolvent operators by the series of Laguerre polynomials. Finally they apply their results to concrete examples: translation, convolution and multiplication semigroups in Lebesgue space. Using holomorphic semigroups they improve previous results.