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In this paper we study approximation theorems for $L^2$-space on Damek-Riccispaces. We prove direct Jackson theorem of approximations for the modulus ofsmoothness defined using spherical mean operator on Damek-Ricci spaces. We alsoprove Nikolskii-Stechkin inequality. To prove these inequalities we usefunctions of bounded spectrum as a tool of approximation. Finally, as anapplication, we prove equivalence of the $K$-functional and modulus ofsmoothness for Damek-Ricci spaces.

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