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

TL;DR

This paper establishes approximation theorems on Damek-Ricci spaces, including Jackson and Nikolskii-Stechkin inequalities, and demonstrates the equivalence of K-functional and modulus of smoothness in this context.

Contribution

It introduces new approximation inequalities and proves the equivalence of key smoothness measures specifically for Damek-Ricci spaces.

Findings

Proved direct Jackson theorem for Damek-Ricci spaces.

Established Nikolskii-Stechkin inequality on Damek-Ricci spaces.

Showed the equivalence of K-functional and modulus of smoothness.

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.