Description of the lipschitz and zygmund classes in terms of the modulus of continuity

Abstract

Let C be the space of 2π-periodic continuous real-valued functions, let $$\begin{gathered} \omega _1 (f,h) = \mathop {\sup }\limits_{0 \leqslant t \leqslant h,x \in \mathbb{R}} |f(x + t/2) - f(x - t/2)|, \hfill \\ \omega _2 (f,h) = \mathop {\sup }\limits_{0 \leqslant t \leqslant h,x \in \mathbb{R}} |f(x - t) - 2f(x) + f(x + t)| \hfill \\ \end{gathered} $$ be first- and second-order moduli of continuity of a function f∈C with step h≥0. Denote by Lip1 = {f ∈ C: ω1(f,h) = O(h)} the Lipschitz class and by Z1 = {f ∈ C: ω2(f,h) = O(h)} the Zygmund class. The class of functions W⊂C is said to be described in terms of the kth modulus of continuity if for any functions f1, f2∈C such that ωk(f2) from f1∈W it follows that f2∈W. As is shown, the class Z1 is not described in terms of the first-order modulus of continuity, whereas the class Lip is not described in terms of the second-order modulus of continuity. Bibliography: 3 titles.