Pseudo prolate spheroidal functions

TL;DR

This paper introduces pseudo prolate spheroidal functions, a new class of functions that approximate eigenfunctions of a time-frequency limiting operator, with a focus on their asymptotic count and explicit construction.

Contribution

It provides a novel approach to counting functions with approximate eigenvalues, leading to explicit construction methods for pseudo prolate spheroidal functions.

Findings

Asymptotic count of pseudo prolate spheroidal functions is approximately (1-ε)^{-1} times the classical estimate.

Explicit construction of pseudo prolate spheroidal functions is demonstrated.

The approach generalizes the classical eigenfunction framework to ε-pseudoeigenfunctions.

Abstract

Let $D_{T}$ and $B_{\Omega }$ denote the operators which cut the time content outside $T$ and the frequency content outside $\Omega $, respectively. The prolate spheroidal functions are the eigenfunctions of the operator $P_{T,\Omega }=D_{T}B_{\Omega }D_{T}$. With the aim of formulating in precise mathematical terms the notion of Nyquist rate, Landau and Pollack have shown that, asymptotically, the number of such functions with eigenvalue close to one is $\approx \frac{\left\vert T\right\vert \left\vert \Omega \right\vert }{2\pi }$. We have recently revisited this problem with a new approach: instead of counting the number of eigenfunctions with eigenvalue close to one, we count the maximum number of orthogonal $\epsilon$-pseudoeigenfunctions with $\epsilon $-pseudoeigenvalue one. Precisely, we count how many orthogonal functions have a maximum of energy $\epsilon $ outside the domain $T\times \Omega $, in the sense that $\left\Vert P_{T,\Omega }f-f\right\Vert ^{2}\leq \epsilon $. We have recently discovered that the sharp asymptotic number is $\approx (1-\epsilon )^{-1}\frac{\left\vert T\right\vert \left\vert \Omega \right\vert }{2\pi }$. The proof involves an explicit construction of the pseudoeigenfunctions of $P_{T,\Omega }$. When $T$ and $\Omega$ are intervals we call them pseudo prolate spheroidal functions. In this paper we explain how they are constructed.