In this paper, we discuss various basic properties of moduli of smoothness of functions from $L_p\left({\mathbb{R}}^d\right)$, $0<p\le \infty$. 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.

中文翻译：

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光滑模量的性质 ${\mathrm{大号}}_p（{\mathbb{[R}}^d）$

在本文中，我们讨论了函数光滑度模量的各种基本性质， ${\mathrm{大号}}_p（{\mathbb{[R}}^d）$， $0<p\le \infty$。特别是，给出了杰克逊型，马尔凯型和乌里扬诺夫型不等式的完整版本。$p$。此外，光滑度的模量与相应的$ķ$-功能和实现概念得到证明。