2017
Том 69
№ 9

All Issues

Smoothness of functions in the metric spaces Lψ

Pichugov S. A.

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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 $\text{Lip}(1, \psi)$.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 9, pp 1382-1402.

Citation Example: Pichugov S. A. Smoothness of functions in the metric spaces Lψ // Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1214-1232.

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