Pseudo prolate spheroidal functions

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 eigen-functions of the operator P-T,P-Omega = DTB Omega DT. 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 approximate to vertical bar T vertical bar vertical bar Omega vertical bar/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 c-pseudoeigenfunctions with epsilon-pseudoeigenvalue one. Precisely, we count how many orthogonal functions have a maximum of energy outside the domain T x Omega, in the sense that parallel to P-T,P-Omega f - f parallel to(2) <= epsilon. We have recently discovered that the sharp asymptotic number is approximate to(1 - epsilon)-1- vertical bar T vertical bar vertical bar Omega vertical bar/2 pi. The proof involves an explicit construction of the pseudoeigenfunctions of P-T,P-Omega. When T and Omega are intervals we call them pseudo prolate spheroidal functions. In this paper we explain how they are constructed.