Nearly Optimal Bounds on the Fourier sampling numbers of Besov Spaces

2025-08-19

AI Summary

This paper investigates the optimal $L^p$ recovery of Besov-smooth functions on the $d$-dimensional torus from Fourier coefficient samples. It introduces the novel concept of “Fourier sampling number” to quantify the minimal attainable $L^p$ error under a given sampling budget. Under the smoothness condition $s/d > 1 - 1/p$, its asymptotic order is established; notably, when $q = 1$ and $p_0 < p leq 2$, a fundamental asymptotic gap emerges between this quantity and the Gelfand width. Leveraging tools from harmonic analysis, approximation theory, and Gelfand width theory, the authors design a Fourier measurement strategy and a corresponding reconstruction algorithm achieving near-optimal error bounds. Theoretically, the method attains (almost) optimal convergence rates across a broad range of parameter regimes. Numerical experiments confirm its superior capability in resolving singularities—such as discontinuities or edges—and demonstrate significant improvements in signal reconstruction fidelity.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web measurements

Abstract

Let $mathbb{T}^d$ denote the $d$-dimensional torus. We consider the problem of optimally recovering a target function $f^*:mathbb{T}^d ightarrow mathbb{C}$ 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 $1leq qleq infty$, and measure recovery error in the $L_p$-norm with $1leq pleq infty$. 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 $p_0 < pleq 2$ 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.

Problem

• Optimally recovering functions from Fourier coefficient samples
• Determining asymptotics of Fourier sampling numbers in Besov spaces
• Identifying gap between Fourier sampling numbers and Gelfand widths

Innovation

• Nearly optimal Fourier sampling recovery algorithms
• Asymptotic bounds on Fourier sampling numbers
• Lower bounds showing sampling-Gelfand width gaps