Efficient Computation of Slepian Functions on the Real Line

In this work, we propose a method for the derivation of prolate spheroidal wave functions (PSWFs) and Slepian functions on continuous and disjoint intervals on the real number line. The proposed method uses Fourier series to obtain a closed-form approximation for Slepian functions on the real line. With this closed-form expression, Slepian functions can be evaluated at arbitrary points in the region of interest with high accuracy. The conventional method uses properties of the Slepian concentration problem to evaluate PSWFs on finite number of points in an interval. The conventional method is computationally expensive and does not allow for easy storage. By approximating an interval containing regions of interest as periodic, we express the Slepian concentration problem as a finite dimensional problem using the Fourier series domain. Solutions to the Slepian concentration problem in this form are Fourier series coefficients corresponding to the Slepian functions. Reconstruction in Fourier series basis, scaling and subsequent truncation provides the closed-form expression for the Slepian problem. Upon comparison with PSWFs obtained by the conventional method, we find negligible difference.