Smoothness of functions in the metric spaces L ψ

Abstract

Let L 0 (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 L ψ and the smoothness in L 1, the relationship between the moduli of continuity of f in L ψ and L 1 and the imbedding theorems for the classes Lip(α, ψ) in L 1, and the structure of functions from the class Lip(1, ψ).