On the estimates of the values of various widths of classes of functions of two variables in the weight space L2,γ (ℝ2), γ = exp(− x2− y2)

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For the classes of functions of two variables W2 (Ωm,γ, Ψ) = {f ∈ L2,γ (ℝ2) : Ωm,γ (f, t) ⩽ Ψ(t) ∀t ∈ (0, 1)}, m ∈ ℕ, where Ωm,γ is a generalized modulus of continuity of the m-th order, and Ψ is a majorant, the upper and lower bounds for the ortho-, Kolmogorov, Bernstein, projective, Gel’fand, and linear widths in the metric of the space L2,γ (ℝ2) are found. The condition for a majorant under which it is possible to calculate the exact values of the listed extreme characteristics of the optimization content is indicated. We consider the similar problem for the classes $$ {W}_2^{r,0} $$ (Ωm,γ, Ψ) = $$ {L}_{2,\upgamma}^{r,0} $$ (D, ℝ2) ∩ $$ {W}_2^r $$ (Ωm,γ, Ψ), r,m ∈ ℕ, ( $$ D=\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {y}^2}-2x\frac{\partial }{\partial x}-2y\frac{\partial }{\partial y} $$ being the differential operator). Those classes consist of functions f ∈ $$ {L}_{2,\upgamma}^{r,0} $$ (ℝ2) whose Fourier–Hermite coefficients are ci0 (f) = c0j (f) = c00 (f) = 0 ∀i, j ∈ ℕ. The r-th iterations Drf = D (Dr−1f) (D0f ≡ f) belong to the space L2,γ (ℝ2) and satisfy the inequality Ωm,γ (Drf, t) ⩽ Ψ(t) ∀t ∈ (0, 1). On the indicated classes, we have determined the upper bounds (including the exact ones) for the Fourier–Hermite coefficients. The exact results obtained are specified, and a number of comments regarding them are given.