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Статья
2021

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


M. O. AkobirshoevM. O. Akobirshoev
Русская математика
https://doi.org/10.3103/S1066369X21090012
Abstract / Full Text

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

$${\mathcal D}:=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}-2x\frac{\partial}{\partial x}-2y\frac{\partial}{\partial y}$$
Author information
  • Technological University of Tajikistan, 63/3 N. Karabaeva Ave., 734055, Dushanbe, Republic of TajikistanM. O. Akobirshoev
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