Mean-Square Approximation by “Angle” in the Space $L_{2,\mu}(\mathbb{R}^{2})$ with the Chebyshev–Hermite Weight

Abstract

Let $L_{2,\mu}(\mathbb{R}^{2}), \ \mu(x,y)=\exp\{-(x^{2}+y^{2})\}, \ \mathbb{R}=(-\infty, +\infty), \ \mathbb{R}^{2}:=\mathbb{R}\times\mathbb{R},$ be the space of functions f, for which $\mu^{1/2}f\in L_{2}(\mathbb{R}^{2}).$ In the metric of space $L_{2,\mu}(\mathbb{R}^{2})$, the sharp inequalities of Jackson–Stechkin type are obtained, which relate the best mean-square approximation by “angle” of functions f from classes $L_{2,\mu}^{r}(\mathbb{R}^{2})$ and the averaged with the weight q generalized mixed modules of continuity $\Omega_{k,l}(D^{r}f)$, where

$${\mathcal D}:=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}-2x\frac{\partial}{\partial x}-2y\frac{\partial}{\partial y}$$

is the second order Chebyshev differential operator.