Pseudo prolate spheroidal functions

Abstract

Abstract-Let. 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 ǫ-pseudoeigenfunctions with ǫ-pseudoeigenvalue one. Precisely, we count how many orthogonal functions have a maximum of energy ǫ outside the domain T × Ω, in the sense that PT,Ωf − f 2 ≤ ǫ.We have recently discovered that the sharp asymptotic number is. The proof involves an explicit construction of the pseudoeigenfunctions of PT,Ω. When T and Ω are intervals we call them pseudo prolate spheroidal functions. In this paper we explain how they are constructed.