Jackson-Type Inequalities for the Special Moduli of Continuity on the Entire Real Axis and the Exact Values of Mean ν - Widths for the Classes of Functions in the Space L2 (ℝ)

The exact values of constants are obtained in the space L2(ℝ) for the Jackson-type inequalities for special moduli of continuity of the k th order defined by the Steklov operator Sh(f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f $$\end{document}) instead of the translation operator Th(f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f $$\end{document}) in the case of approximation by entire functions of exponential type σ ∈ (0,∞) . The exact values of the mean ν -widths (linear, Bernstein, and Kolmogorov) are also obtained for the classes of functions defined by the indicated characteristic of smoothness.