Direct and inverse approximation theorems for functions defined in Damek–Ricci spaces

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UDC 517.5 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 L 2 -space for the Damek–Ricci spaces. These theorems are analogous to the well-known theorems of Jackson and Bernstein. We also consider some problems related to the constructive characteristics of functional classes defined by the majorants of the moduli of smoothness of their elements.