Boas-type theorems for Laguerre type operator

Abstract

Let ${\mathbb {K}}=[0,+\infty )\times {\mathbb {R}}$ the Laguerre Hypergroup. In this paper, an analogous of Boas-type results is established for ${\mathcal {F}}_{L}(f)$, the laguerre transform of f, and we give necessary and sufficient conditions in terms of ${\mathcal {F}}_{L}(f)$, to ensure that f belongs either to one of the generalized Lipschitz classes ${\mathbf {H}}_{\alpha }^{k}({\mathbb {K}})$ and ${\mathbf {h}}_{\alpha }^{k}({\mathbb {K}})$ for $\alpha >0$.