On the estimates of the values of various widths of classes of functions of two variables in the weight space $L_{2,\gamma}(\mathbb{R}^2)$, $\gamma=\exp(-x^2-y^2)$

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 $W^{r,0}_2(\Omega_{m,\gamma},\Psi)=L^{r,0}_{2,\gamma}(D,\mathbb{R}^2) \cap W^r_2(\Omega_{m,\gamma},\Psi)$, $r,m \in \mathbb{N}$, $\big(D=\frac{\displaystyle \partial^2}{\displaystyle \partial x^2} + \frac{\displaystyle \partial^2}{\displaystyle \partial y^2} -2x\frac{\displaystyle \partial}{\displaystyle \partial x} -2y\frac{\displaystyle \partial}{\displaystyle \partial y}$ 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.

Quantitative results for positive linear operators which preserve certain functions

In this paper we obtain estimations of the errors in approximation by positive linear operators which fix certain functions. We use both the first and the second order classical moduli of smoothness and a generalized modulus of continuity of order two. Some applications involving Bernstein type operators, Kantorovich type operators and genuine Bernstein-Durrmeyer type operators are presented.

On approximation by trigonometrical polynomials in $L_2$ space

We obtain certain inequalities of Jackson type, connecting the value of the best approximation of periodic differentiable functions and the generalized modulus of continuity of the highest derivative.

On estimates for the Jacobi transform in the space $L^{p}(\mathbb{R}^{+}, J^{\alpha,\beta}(x)dx)$

In this paper, we prove the estimates for the Jacobi transform in $L^{p}(\mathbb{R}^{+}, J^{\alpha,\beta}(x)dx)$ as applied to some classes of functions characterized by a generalized modulus of continuity.

Inequalities of Jackson-Stechkin type for $B^2$-almost periodic functions

We obtain sharp inequalities of the Jackson-Stechkin type for approximation of $B^2$-almost periodic functions, containing the generalized modulus of continuity.