Spectral Decay of Time and Frequency Limiting Operator

TL;DR

This paper provides an explicit integral approximation for the eigenvalues of the time and frequency limiting operator, revealing their super-exponential decay and enabling efficient computation for large parameters.

Contribution

It introduces a new integral approximation formula for the eigenvalues of the operator, accurately capturing their decay rate and facilitating low-cost computations for large bandwidths and indices.

Findings

Eigenvalues exhibit super-exponential decay starting from the plunge region.

The approximation formula is accurate even for large parameters.

Numerical examples confirm the theoretical results.

Abstract

For fixed $c,$ the Prolate Spheroidal Wave Functions (PSWFs) $\psi_{n, c}$ form a basis with remarkable properties for the space of band-limited functions with bandwidth $c$. They have been largely studied and used after the seminal work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications rely heavily of the behavior and the decay rate of the eigenvalues $(\lambda_n(c))_{n\geq 0}$ of the time and frequency limiting operator, which we denote by $\mathcal Q_c.$ Hence, the issue of the accurate estimation of the spectrum of this operator has attracted a considerable interest, both in numerical and theoretical studies. In this work, we give an explicit integral approximation formula for these eigenvalues. This approximation holds true starting from the plunge region where the spectrum of $\mathcal Q_c$ starts to have a fast decay. As a consequence of our explicit approximation formula, we give a precise description of the super-exponential decay rate of the $\lambda_n(c).$ Also, we mention that the described approximation scheme provides us with fairly accurate approximations of the $\lambda_n(c)$ with low computational load, even for very large values of the parameters $c$ and $n.$ Finally, we provide the reader with some numerical examples that illustrate the different results of this work.