Widths of the Classes of Functions in the Weight Space L2,γ(ℝ), γ = exp(−X2)

In the space L2,γ(ℝ), we study the approximating optimization characteristics for the function classes W2rΩm,γ,φΨℝ≔f∈L2,γrDℝ∫0tΩm,γDrfuφudu≤Ψt∀t∈01,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {W}_{2}^{r}\left({\Omega}_{m,\upgamma,}\varphi, \Psi; \mathbb{R}\right):= \left\{f\in {L}_{2,\upgamma}^{r}\left(D,\mathbb{R}\right);\kern1em {\int}_{0}^{t}{\Omega}_{m,\upgamma}\left({D}^{r}f,u\right)\varphi (u) du\le \Psi (t)\forall t\in \left(0,1\right)\right\}, $$\end{document} where r ∈ ℤ+, m ∈ ℕ, Ωm,γ is the generalized modulus of continuity of order m, 𝜑 is a weight function, is a majorant, D≔−d2dx2+2xddx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D\kern0.5em := -\frac{d^2}{dx^2}+2x\frac{d}{dx} $$\end{document} is a differential operator, Drf = D(Dr−1f) (r ∈ ℕ), D0f ≡ f, and L02,γ(D, ℝ) ≡ L2,γ(ℝ). For various widths of the indicated classes in L2,γ(ℝ), we establish their lower and upper estimates and present the conditions for the majorant under which the exact values of these widths can be determined. Some specific exact results are also presented.