The Diffraction Limit and Extremal Functions

It is widely believed that the physics of diffraction imposes certain fundamental limits on the resolution of an optical system. In this work we study the diffraction limit as a statistical inverse problem in increasingly more realistic mathematical models and prove rigorous upper and lower bounds on the sample complexity of resolving closely-spaced point sources. Our techniques center around extremal functions, namely the Beurling-Selberg majorant and its higher-dimensional generalizations, which we use to show the emergence of a sharp phase transition that governs when it is possible to recover fine-grained structure from low-frequency measurements.

Based on joint work with Sitan Chen.