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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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Abstract

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.

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Authors and Affiliations

  1. A. Nobel University, Dnepr, Ukraine

    Sergey B. Vakarchuk

  2. Oles Honchar Dnepr National University, Dnepr, Ukraine

    Mihail B. Vakarchuk

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  1. Sergey B. Vakarchuk
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  2. Mihail B. Vakarchuk
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Correspondence to Sergey B. Vakarchuk.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 1, pp. 94–102 January–March, 2020.

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Vakarchuk, S.B., Vakarchuk, M.B. 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). J Math Sci 248, 217–232 (2020). https://doi.org/10.1007/s10958-020-04871-5

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