Smoothness of functions in the metric spaces Lψ - Pichugov, S. A. (Ukrains’kyi Matematychnyi Zhurnal, 2012)
Let $L_0(T)$ be thе set of real-valued periodic measurable functions, let $ψ : R^+ → R^+$ be a modulus of continuity $(ψ eq 0)$ , and let$$L_{ψ} \equiv L_{ψ}(T ) = \left\{f \in L_0 (T ): ||f||_{ψ} := \int_T ψ( |f (x)| ) dx < ∞ \right\}.$$The following problems are investigated:Relationship between the rate of approximation of $f$ by trigonometric polynomials in $L_{ψ}$ and smoothnessin $L_1$.Correlation between the moduli of continuity of $f$ in $L_{ψ}$ and $L_1$, and theorems on imbedding of the classes $\text{Lip} (α, ψ)$ in $L_1$.Structure of functions from the class $\text{Lip}(1, ψ)$.