Best $m$-term trigonometric approximation in weighted Wiener spaces and applications
Moritz Moeller, Serhii Stasyuk, Tino Ullrich
TL;DR
The paper analyzes sharp asymptotics for best $m$-term trigonometric approximation in weighted Wiener spaces $S^{r}_{\theta}\mathcal{A}$ with mixed smoothness, and investigates how these widths control nonlinear sampling recovery in multivariate function spaces. Central to the approach are a frequency-layer decomposition, embeddings into (unweighted) Wiener spaces, and instance-optimal recovery results that bound $L_q$-recovery errors by sums of best $m$-term and trigonometric approximation errors. The authors establish precise asymptotics for $\sigma_m(S^{r}_{\theta}\mathcal{A})_{L_q}$ and related $\mathcal{A}$-norm widths, along with a suite of embeddings linking weighted Wiener spaces to Besov-Sobolev spaces with bounded mixed derivatives, enabling transfer of width and sampling-recovery bounds. They further derive nonlinear sampling-width bounds for weighted Wiener and Besov-Sobolev spaces, showing regime-dependent gains of nonlinear methods (e.g., instance-optimal recovery) over linear approaches and highlighting the role of the de la Vallée Poussin operator in constructive bounds. Collectively, the results advance understanding of multivariate approximation in mixed-smoothness spaces and provide tools for effective recovery from samples in high dimensions.
Abstract
In this paper we study best \(m\)-term trigonometric approximation in weighted Wiener spaces and its consequences for Besov and Sobolev spaces with bounded mixed derivative/difference. We obtain several sharp asymptotic bounds for weighted Wiener spaces including the quasi-Banach case. It has recently been observed that best \(m\)-term trigonometric widths in the uniform norm together with recovery algorithms stemming from compressed sensing serve to control the optimal sampling recovery error in various relevant spaces of multivariate functions. We use a collection of old and new tools as well as novel findings to extend the recovery bounds to classical multivariate smoothness spaces. It turns out that embeddings into Wiener spaces serve as a powerful tool to improve certain recent bounds.