# Widths of classes of functions in the weight space $L_{2,\gamma}(\mathbb{R}), \gamma=\exp(-x^2)$

• S. B. Vakarchuk Alfred Nobel University, Dnipro
Keywords: the best polynomial approximation, series Fourier - Hermite, width

### Abstract

UDC 517.5

In the space $L_{2,\gamma}(\mathbb{R})$ approximating characteristics of the optimizing sense have been considered for the classes $W^r_2(\Omega_{m,\gamma}, \varphi,\Psi; \mathbb{R}) :=$ $:= \Big\{ f \in L^r_{2,\gamma}(D,\mathbb{R}) : \int\limits_0^t \Omega_{m,\gamma} (D^rf, u) \varphi(u) du \leqslant \Psi(t) \, \forall t \in (0,1) \Big\}$, where $r \in \mathbb{Z}_{+}$; $m \in \mathbb{N}$; $\Omega_{m,\gamma}$ is the generalized $m$th order modulus of continuity; $\varphi$ is a weight function; $\Psi$ is a majorant; $D := - \fracd^2}d x^2} +2x \fracd}d x$ is the differential operator, $D^r f = D(D^{r-1} f)$ $(r \in \mathbb{N})$, $D^0 f \equiv f$; $L^0_{2,\gamma}(D,\mathbb{R}) \equiv L_{2,\gamma}(\mathbb{R})$. Lower and upper estimates were found for the different widths of the indicated classes in $L_{2,\gamma}(\mathbb{R})$. The conditions on the majorant have been determined under which realization their exact values succeed to compute. Some concrete exact rezults given.

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Published
17.06.2022
How to Cite
Vakarchuk, S. B. “Widths of Classes of Functions in the Weight Space $L_{2,\gamma}(\mathbb{R}), \gamma=\exp(-x^2)$”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 5, June 2022, pp. 610 -19, doi:10.37863/umzh.v74i5.7147.
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Research articles