Mean-square approximation by “angle” in the space $L_{2,\mu}(\mathbb{R}^{2})$ with the Chebyshev–Hermite weight

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 which relate the best mean squared approximation by “angle” formed with an algebraic polynomials of two variables averaged with Chebyshev-Hermite weight $L_{\nu} \ (1\leq \nu\leq\infty)$ and norm of module of continuity $k$-th order by variable $x$ and $l$-th order by variable $y$ with derivatives ${\mathcal D}^{r}f,$ were obtained. ${\mathcal D}$ — is Chebyshev differential operator of second order of form
$${\mathcal D}:=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}-2x\frac{\partial}{\partial x}-2y\frac{\partial}{\partial y}.$$