Journal of Mathematical Sciences

, Volume 97, Issue 4, pp 4233–4237 | Cite as

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

  • O. L. Vinogradov


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.


Lipschitz Class Zygmund Class 
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  1. 1.
    I. P. Natanson,Constructive Function Theory Vols. 1–3, Frederick Ungar, New York (1964, 1965, 1965).Google Scholar
  2. 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. 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

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© Kluwer Academic/Plenum Publishers 1999

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  • O. L. Vinogradov

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