Generalized Prolate Spheroidal Wave Functions Associated With Linear Canonical Transform

Time-limited and (a, b, c, d)-band-limited signals are of great interest not only in theory but also in real applications. In this paper, we use the sampling theorem associated with linear canonical transform to investigate an operator whose effect on a signal is to produce its first time-limited then (b, b, c, d)-band-limited version. First, the eigenvalue problem for the operator is shown to be equivalent to a discrete eigenvalue problem for an infinite matrix. Then the eigenfunctions of the operator, which are referred to as generalized prolate spheroidal wave functions (GPSWFs), are shown to be first, orthogonal over finite as well as infinite intervals, and second, complete over L
²
(-L, L) and the class of (a, b, c, d)-band-limited signals. A simple method based on sampling theorem for computing GPSWFs is presented and the definite parity of GPSWFs is also given. Finally, based on the dual orthogonality and completeness of GPSWFs, several applications of GPSWFs to the representation of time-limited and (a, b, c, d)-band-limited signals are presented.