Uniform bounds of prolate spheroidal wave functions and eigenvalues decay

Abstract

The prolate spheroidal wave functions (PSWFs) form a set of special functions with remarkable properties. They are defined on [−1,1] as the bounded eigenfunctions ψn,c of a Sturm–Liouville differential operator Lc as well as the eigenfunctions of the linear integral operator Qc with kernel sin(c(x−y))π(x−y). We give new bounds for the values ψn,c(0), ψn,c′(0) and ψn,c(1), which allow us to obtain estimates for the Lp norms of the PSWFs and for eigenvalues of Lc and Qc. We get in particular an almost sharp exponential lower decay rate of the eigenvalues of Qc.