Smoothness of functions in the metric spaces Lψ

Abstract

Let $L_0(T)$ be thе set of real-valued periodic measurable functions, let $\psi : R^+ \rightarrow R^+$ be a modulus of continuity $(\psi \neq 0)$ , and let $$L_{\psi} \equiv L_{\psi}(T ) = \left\{f \in L_0 (T ): ||f||_{\psi} := \int_T \psi( |f (x)| ) dx < \infty \right\}.$$ The following problems are investigated: Relationship between the rate of approximation of $f$ by trigonometric polynomials in $L_{\psi}$ and smoothness in $L_1$. Correlation between the moduli of continuity of $f$ in $L_{\psi}$ and $L_1$, and theorems on imbedding of the classes $\text{Lip} (\alpha, \psi)$ in $L_1$. Structure of functions from the class $\text{Lip}(1, \psi)$.