  In this work, we first give some mathematical preliminaries concerning the
generalized prolate spheroidal wave function (GPSWFs). These set of special
functions have been introduced in [16] and [7] and they are defined as the
infinite and countable set of the eigenfunctions of a weighted finite Fourier
transform operator. Then, we show that the set of the singular values of this
operator decay at a super-exponential decay rate. We also give some local
estimates and bounds of these GPSWFs. As a first application of the spectral
properties of the GPSWFs and their associated eigenvalues, we give their
quality of approximation in a periodic Sobolev space. Our second application is
related to a generalization in the context of the GPSWFs, of the Landau-Pollak
approximate dimension of the space of band-limited and almost time-limited
functions, given in the context of the classical PSWFs, the eigenfunctions of
the finite Fourier transform operator. Finally, we provide the reader with some
numerical examples that illustrate the different results of this work.
