Pseudo prolate spheroidal functions

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.