Smoothness of functions in the metric spaces Lψ

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {L_{\uppsi }}\equiv {L_{\uppsi }}(T)=\left\{ {f\in {L_0}(T):{{{\left\| f \right\|}}_{\uppsi }}:=\int\limits_T {\uppsi \left( {\left| {f(x)} \right|} \right)dx<\infty } } \right\}. $$\end{document}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).