Properties of moduli of smoothness in Lp(Rd)

Abstract

In this paper, we discuss various basic properties of moduli of smoothness of functions from Lp(Rd), 0<p≤∞. In particular, complete versions of Jackson-, Marchaud-, and Ulyanov-type inequalities are given for the whole range of p. Moreover, equivalences between moduli of smoothness and the corresponding K-functionals and the realization concept are proved. © 2020 Elsevier Inc.