On Spectral Approximations by Generalized Slepian Functions

Abstract

We introduce a family of orthogonal functions, termed as generalized Slepian functions (GSFs),  closely related to the time-frequency concentration problem on a unit disk in D. Slepian [19]. These functions form a complete orthogonal system in $L^2_{\varpi_α}(-1,1)$ with $\varpi_α=(1-x)^α,$ $α>-1,$ and can be viewed as a generalization of the Jacobi polynomials with parameter $(\alpha,0)$. We present various analytic and asymptotic properties of GSFs, and study spectral approximations by such functions.