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 L_ψ(T ) = {f L_0 (T ): ||f||_ψ := _T ψ( |f (x)| ) dx < ∞ }. The following problems are investigated: Relationship between the rate of approximation of f by trigonometric polynomials in L_ψ and smoothness in L_1. Correlation between the moduli of continuity of f in L_ψ and L_1, and theorems on imbedding of the classes Lip (α, ψ) in L_1. Structure of functions from the class Lip(1, ψ).