Jackson’s inequalities in Laguerre hypergroup

Abstract

In this work, we consider the Laguerre hypergroup $(K, *_{\alpha })$, where ${\mathbb {K}} = [0,+\infty [ \times {\mathbb {R}}$ and $*_{\alpha }$ a convolution product on ${\mathbb {K}}$ coming from the product formula satisfied by the Laguerre functions ${\mathcal {L}}_{m}^{(\alpha )}$ $(m\in {\mathbb {N}}, \alpha \ge 0)$. We give new estimates for the Laguerre kernel $\varphi _{\lambda ,m}(x,t)$, $(x,t)\in {\mathbb {K}}$, $(\lambda ,m)\in {\mathbb {R}}\times {\mathbb {N}}$. Analogues of Jackson’s theorems useful in applications are proved for the Fourier–Laguerre transform in the space $L^{2}_{\alpha }({\mathbb {K}})$.