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 L2̟α (−1,1) with ̟α(x) = (1 − x)α, α> −1, and can be viewed as a generalization of the Jacobi polynomials with parameter (α, 0). We present various analytic and asymptotic properties of GSFs, and study spectral approximations by such functions