Weighted eigenfunction estimates with applications to compressed sensing

TL;DR

This paper develops weighted eigenfunction estimates for convex surfaces of revolution using semiclassical analysis, leading to improved sampling bounds for sparse eigenfunction recovery, especially on the sphere.

Contribution

It introduces new weighted eigenfunction estimates and applies them to derive enhanced sampling bounds for sparse eigenfunction expansions on surfaces of revolution.

Findings

Sparse eigenfunction expansions can be recovered with fewer samples than previously known.

On the sphere, functions with s-sparse expansions are recoverable from m > s N^(1/6) log^4(N) samples.

The estimates improve understanding of sampling requirements in eigenfunction-based signal processing.

Abstract

Using tools from semiclassical analysis, we give weighted L^\infty estimates for eigenfunctions of strictly convex surfaces of revolution. These estimates give rise to new sampling techniques and provide improved bounds on the number of samples necessary for recovering sparse eigenfunction expansions on surfaces of revolution. On the sphere, our estimates imply that any function having an s-sparse expansion in the first N spherical harmonics can be efficiently recovered from its values at m > s N^(1/6) log^4(N) sampling points.