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Properties of moduli of smoothness in $L_p(\mathbb{R}^d)$ (1907.12788v3)
Published 30 Jul 2019 in math.CA
Abstract: In this paper, we discuss various basic properties of moduli of smoothness of functions from $L_p(\mathbb{R}d)$, $0<p\le \infty$. 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.