Further Spectral Properties of the Weighted Finite Fourier Transform Operator and Related Applications

TL;DR

This paper investigates the spectral properties of the weighted finite Fourier transform operator, focusing on the decay of its singular values, bounds of GPSWFs, and their applications in approximation theory and dimension estimation.

Contribution

It provides new insights into the decay rates of singular values and extends the Landau-Pollak dimension concept to GPSWFs, with numerical illustrations.

Findings

Singular values decay super-exponentially.

GPSWFs have bounded local estimates.

Extended Landau-Pollak dimension for GPSWFs.

Abstract

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.