Weighted sampling recovery of functions with mixed smoothness

TL;DR

This paper investigates the optimality of linear weighted sampling algorithms for approximating functions with mixed smoothness in weighted Sobolev spaces, providing constructions on sparse grids and establishing convergence rates.

Contribution

It introduces new linear sampling algorithms on sparse grids for functions with mixed smoothness and proves their optimality in terms of sampling widths.

Findings

Algorithms achieve optimal convergence rates in 1D.

Sparse grid sampling provides effective approximation bounds.

Theoretical analysis of sampling widths confirms optimality.

Abstract

We studied linear weighted sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on $\mathbb{R}^d$ from a set of $n$ their sampled values. Functions to be recovered are in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness, and the approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$. Here, the weight $w$ is a tensor-product Freud-type weight. The optimality of linear sampling algorithms is investigated in terms of sampling $n$-widths. We constructed linear sampling algorithms on sparse grids of sampled points which form a step hyperbolic cross in the function domain, and which give upper bounds for the corresponding sampling $n$-widths. We proved that in the one-dimensional case, these algorithms realize the exact convergence rate of the $n$-sampling widths.