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

Article

- 136 Downloads

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.I. P. Natanson,
*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:
*Mathematics in the USSR over the Thirty Years*1917–1947 [in Russian], Moscow-Leningrad (1948), pp. 288–318.Google Scholar

## Copyright information

© Kluwer Academic/Plenum Publishers 1999