Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in L2

We find the exact values of the n-widths for the classes of periodic differentiable functions in L2[0, 2π] satisfying the constraint \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int\limits_0^h {t\tilde \Omega _m^{1/m} (f^{(r)} ;t)dt \leqslant \Phi (h)} ,$\end{document} where h > 0, m ∈ ℕ, r ∈ ℤ+, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde \Omega _m $\end{document}(f(r); t) is the generalized mth order continuity modulus of the derivative f(r) ∈ L2[0, 2π], while Φ(t) is an arbitrary increasing function such that Φ(0) = 0.