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 $W^r_{p,w}(\mathbb{R})$. 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 $Q_{\rho,m}$ and $P_{\rho,m}$ which are asymptotically optimal in terms of the sampling $n$-widths $\varrho_n(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R}))$ for every pair $p,q \in [1,\infty]$, and prove the right convergence rate of these sampling $n$-widths, where $\boldsymbol{W}^r_{p,w}(\mathbb{R})$ denotes the unit ball in $W^r_{p,w}(\mathbb{R})$. The algorithms $Q_{\rho,m}$ and $P_{\rho,m}$ 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 $Q_{\rho,m}$ and $P_{\rho,m}$, for Freud-weighted numerical integration of functions in $W^r_{p,w}(\mathbb{R})$.