A note on $K$-functional, Modulus of smoothness, Jackson theorem and Nikolskii-Stechkin inequality on Damek-Ricci spaces

Abstract

In this paper we study approximation theorems for L 2 -space on Damek-Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek-Ricci spaces. We also prove Nikolskii-Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we prove equivalence of the K-functional and modulus of smoothness for Damek-Ricci spaces. Contents 1. Introduction 2. Essentials about harmonic NA groups 3. Main results 3.1. Direct Jackson theorem 3.2. Nikolskii-Stechkin inequality