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- W3081465502 abstract "For the classes of functions of two variables $W_2(Omega_{m,gamma},Psi)={ f in L_{2,gamma}(mathbb{R}^2) : Omega_{m,gamma}(f,t) leqslant Psi(t) , forall t in (0,1)}$, $m in mathbb{N}$, where $Omega_{m,gamma}$ is a generalized modulus of continuity of the $m$-th order, and $Psi$ 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 $L_{2,gamma}(mathbb{R}^2)$ 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 $W^{r,0}_2(Omega_{m,gamma},Psi)=L^{r,0}_{2,gamma}(D,mathbb{R}^2) cap W^r_2(Omega_{m,gamma},Psi)$, $r,m in mathbb{N}$, $big(D=frac{displaystyle partial^2}{displaystyle partial x^2} + frac{displaystyle partial^2}{displaystyle partial y^2} -2xfrac{displaystyle partial}{displaystyle partial x} -2yfrac{displaystyle partial}{displaystyle partial y}$ being the differential operator$big)$. Those classes consist of functions $f in L^{r,0}_{2,gamma}(mathbb{R}^2)$ whose Fourier--Hermite coefficients are $c_{i0}(f) = c_{0j}(f)=c_{00}(f)=0$ $forall i, j in mathbb{N}$. The $r$-th iterations $D^rf = D(D^{r-1}f)$ $(D^0f equiv f)$ belong to the space $L_{2,gamma}(mathbb{R}^2)$ and satisfy the inequality $Omega_{m,gamma}(D^rf,t) leqslant Psi(t)$ $forall t in (0,1)$. 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." @default.
- W3081465502 created "2020-09-01" @default.
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- W3081465502 date "2020-03-31" @default.
- W3081465502 modified "2023-10-03" @default.
- W3081465502 title "On the estimates of the values of various widths of classes of functions of two variables in the weight space $L_{2,gamma}(mathbb{R}^2)$, $gamma=exp(-x^2-y^2)$" @default.
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- W3081465502 doi "https://doi.org/10.37069/1810-3200-2020-17-1-7" @default.
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