Prolate Spheroidal Wave Functions and the Accuracy and Dimensionality of Spectral Analysis

TL;DR

This thesis introduces an efficient protocol for high-precision frequency determination of signals using prolate spheroidal wave functions, with new bounds and approximation theory for spectral analysis.

Contribution

It develops a fundamental precision guarantee for spectral analysis and extends the theory of PSWFs to improve truncation estimates and provide new geometric insights.

Findings

Established a high-precision frequency determination protocol.

Extended PSWF concentration bounds for improved spectral analysis.

Provided new geometric insights into operator commutation relations.

Abstract

The main result of this thesis is an efficient protocol to determine the frequencies of a signal $C(t)= \sum_k |a_k|^2 e^{i \omega_k t}$, which is given for a finite time, to a high degree of precision. Specifically, we develop a theorem that provides a fundamental precision guarantee. Additionally, we establish an approximation theory for spectral analysis through low-dimensional subspaces that can be applied to a wide range of problems. The signal processing routine relies on a symmetry between harmonic analysis and quantum mechanics. In this context, prolate spheroidal wave functions (PSWF) are identified as the optimal information processing basis. To establish rigorous precision guarantees, we extend the concentration properties of PSWFs to a supremum bound and an $\ell_2$ bound on their derivatives. The new bounds allow us to refine the truncation estimates for the prolate sampling formula. We also provide a new geometrical insight into the commutation relation between an integral operator and a differential operator, both of which have PSWFs as eigenfunctions.