A note of some approximation theorems of functions on the Laguerre hypergroup

Abstract

This paper uses some basic notions and results on the Laguerre hypergroup K = [0, +?)xR to study some problems in the theory of approximation of functions in the space L2 ?(K). Analogues of the direct Jackson theorems of approximations for the modulus of smoothness (of arbitrary order) constructed by using the generalized translation operators on K are proved. The Nikolskii-Stechkin inequality is also obtained. In conclusion of this work, we show that the modulus of smoothness and the K-functionals constructed from the Sobolev-type space corresponding to the Laguerre operator L? are equivalent.