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

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## Abstract

Let C be the space of 2π-periodic continuous real-valued functions, let be first- and second-order moduli of continuity of a function f∈C with step h≥0. Denote by Lip

$$\begin{gathered} \omega _1 (f,h) = \mathop {\sup }\limits_{0 \leqslant t \leqslant h,x \in \mathbb{R}} |f(x + t/2) - f(x - t/2)|, \hfill \\ \omega _2 (f,h) = \mathop {\sup }\limits_{0 \leqslant t \leqslant h,x \in \mathbb{R}} |f(x - t) - 2f(x) + f(x + t)| \hfill \\ \end{gathered} $$

_{1}= {f ∈ C: ω_{1}(f,h) = O(h)} the Lipschitz class and by Z_{1}= {f ∈ C: ω_{2}(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 f_{1}, f_{2}∈C such that ω_{k}(f_{2}) from f_{1}∈W it follows that f_{2}∈W. As is shown, the class Z_{1}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.### Keywords

Lipschitz Class Zygmund Class## Preview

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### References

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*Constructive Function Theory*Vols. 1–3, Frederick Ungar, New York (1964, 1965, 1965).Google Scholar - 2.S. N. Bernshtein, “On the best approximation of continuous functions by polynomials of given degree,” in:
*Collected Works*[in Russian], Vol. 1, Izd. Akad. Nauk SSSR, Moscow (1952), pp. 11–104.Google Scholar - 3.S. M. Nikol'skii, “Approximation of functions of real variables by polynomials,” in:
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© Kluwer Academic/Plenum Publishers 1999