Ukrainian Mathematical Journal

, Volume 64, Issue 9, pp 1382–1402 | Cite as

Smoothness of functions in the metric spaces Lψ

  • S. A. Pichugov
Article
Let L0 (T ) be the set of real-valued periodic measurable functions, let ψ : R+R+ be a modulus of continuity (ψ ≠ 0) , and let
$$ {L_{\uppsi }}\equiv {L_{\uppsi }}(T)=\left\{ {f\in {L_0}(T):{{{\left\| f \right\|}}_{\uppsi }}:=\int\limits_T {\uppsi \left( {\left| {f(x)} \right|} \right)dx<\infty } } \right\}. $$
The following problems are investigated: the relationship between the rate of approximation of f by trigonometric polynomials in Lψ and the smoothness in L1, the relationship between the moduli of continuity of f in Lψ and L1 and the imbedding theorems for the classes Lip(α, ψ) in L1, and the structure of functions from the class Lip(1, ψ).

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • S. A. Pichugov
    • 1
  1. 1.Dnepropetrovsk National University of Railway TransportDnepropetrovskUkraine

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