Direct and Inverse Approximation Theorems for Functions Defined in Damek–Ricci Spaces

We introduce the notion of k th modulus of smoothness and establish the direct and inverse theorems in terms of the quantities E_s(f) and the moduli of smoothness generated by the spherical mean operator defined on the L²-space for the Damek–Ricci spaces. These theorems are analogous to the well-known Jackson and Bernstein theorems. We also consider some problems related to the constructive characteristics of the functional classes defined by the majorants of the moduli of smoothness of their elements.