Convergence of Prolate Spheroidal Wavelets in a Generalized Sobolev Space and Frames

Abstract

It has been noticed that prolate spheroidal wave functions and associated wavelets have many remarkable properties leading to new applications in electrical engineering and mathematics. In this paper we have studied the modiï¬ed wavelets at different scales to retain a constant energy concentration interval and shown that the sequence of these wavelet bases form a frame (more general than Riesz basis) for Vm (Paley -Wiener spaces in which the sinc function replaced by prolate spheroidal wave functions from wavelet basis). Also, the convergence of associated approximations have been studied in generalized sobolev space which contains the Schwartz space as a particular case and our space is generalized the spaces studied by Pathak [15] and Hormander [9].