Weighted sampling recovery of functions with mixed smoothness
Dinh Dũng
TL;DR
This work addresses optimal linear weighted sampling recovery for functions with mixed smoothness on ${\bf R}^d$ in weighted Sobolev spaces ${W}^r_{p,w}({\bf R}^d)$ with Freud-type tensor-product weights, measuring error in ${L}_{q,w}({\bf R}^d)$. The authors develop sparse-grid linear sampling algorithms on step hyperbolic-cross grids and obtain upper bounds for sampling widths ${\boldsymbol{\rho}}_n({\boldsymbol{W}}^r_{p,w}({\bf R}^d), {L}_{q,w}({\bf R}^d))$, proving exact rates in one dimension and explicit bound forms in higher dimensions. The univariate analysis yields the exact rate ${\boldsymbol{\rho}}_n( {\boldsymbol{W}}^r_{p,w}({\bf R}), {L}_{q,w}({\bf R})) \asymp n^{-r_{\text{λ},p,q}}$ for $1<p<\infty$, while the multivariate extension provides upper and lower bounds with logarithmic factors depending on $(p,q, d)$. These results advance understanding of optimal sampling strategies for weighted mixed-smoothness function classes and illuminate the impact of weights and dimension on attainable recovery rates.
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.