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

M. O. Akobirshoev

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2021), pp. 3-12

Résumé

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

Détail

Keywords: the best approximation with “angle”, translation operator, weight function, Chebyshev-Hermite operator, generalized module of continuity.