Weighted approximate sampling recovery and integration based on B-spline interpolation and quasi-interpolation

Abstract

We propose novel methods for approximate sampling recovery and integration of functions in the Freud-weighted Sobolev space $W_{p,w}^r(\mathbb{R})$. The approximation error of sampling recovery is measured in the norm of the Freud-weighted Lebesgue space $L_{q,w}(\mathbb{R})$. Namely, we construct equidistant compact-supported B-spline quasi-interpolation and interpolation sampling algorithms $Q_{\ddot{I},m}$ and $P_{\ddot{I},m}$ which are asymptotically optimal in terms of the sampling $n$-widths ${\varrho }_n({\mathbf{W}}_{p,w}^r(\mathbb{R}),L_{q,w}(\mathbb{R}))$ for every pair $p,q\in [1,\infty ]$, and prove the right convergence rate of these sampling $n$-widths, where ${\mathbf{W}}_{p,w}^r(\mathbb{R})$ denotes the unit ball in $W_{p,w}^r(\mathbb{R})$. The algorithms $Q_{\ddot{I},m}$ and $P_{\ddot{I},m}$ are based on truncated scaled B-spline quasi-interpolation and interpolation, respectively. We also prove the asymptotical optimality and right convergence rate of the equidistant quadratures generated from $Q_{\ddot{I},m}$ and $P_{\ddot{I},m}$, for Freud-weighted numerical integration of functions in $W_{p,w}^r(\mathbb{R})$.