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

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; $D := - \frac{\displaystyle d^2}{\displaystyle d x^2} +2x \frac{\displaystyle d}{\displaystyle d x}$ 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.