Nearly optimal bounds on the Fourier sampling numbers of Besov spaces

TL;DR

This paper characterizes the asymptotic behavior of Fourier sampling numbers for Besov spaces, providing nearly optimal bounds and algorithms for function recovery from Fourier samples, with implications for edge detection.

Contribution

It determines the asymptotics of Fourier sampling numbers for Besov spaces and introduces nearly optimal measurement and recovery methods, including new lower bounds and practical implications.

Findings

Asymptotic bounds for Fourier sampling numbers in certain regimes

Nearly optimal Fourier measurement and recovery algorithms

A novel lower bound showing gaps for specific parameters

Abstract

Let $\mathbb{T}^d$ denote the $d$-dimensional torus. We consider the problem of optimally recovering a target function $f^*:\mathbb{T}^d\rightarrow \mathbb{C}$ from samples of its Fourier coefficients. We make classical smoothness assumptions on $f^*$, specifically that $f^*$ lies in a Besov space $B^s_\infty(L_q)$ with $s > 0$ and $1\leq q\leq \infty$, and measure recovery error in the $L_p$-norm with $1\leq p\leq \infty$. Abstractly, the optimal recovery error is characterized by a `restricted' version of the Gelfand widths, which we call the Fourier sampling numbers. Up to logarithmic factors, we determine the correct asymptotics of the Fourier sampling numbers in the regime $s/d > 1 - 1/p$. We also give a description of nearly optimal Fourier measurements and recovery algorithms in each of these cases. In the other direction, we prove a novel lower bound showing that there is an asymptotic gap between the Fourier sampling numbers and the Gelfand widths when $q = 1$ and $p_0 < p\leq 2$ with $p_0 \approx 1.535$. Finally, we discuss the practical implications of our results, which imply a sharper recovery of edges, and provide numerical results demonstrating this phenomenon.