Weighted finite Fourier transform operator: Uniform approximations of the eigenfunctions, eigenvalues decay and behaviour

TL;DR

This paper provides uniform asymptotic approximations for eigenfunctions of the weighted finite Fourier transform operator, demonstrating super-exponential decay of eigenvalues and bounds on eigenvalue counting for large parameters.

Contribution

It introduces two uniform asymptotic approximations of eigenfunctions using Bessel functions and Jacobi polynomials, and analyzes eigenvalue decay and counting bounds.

Findings

Eigenfunctions approximated by Bessel functions and Jacobi polynomials

Eigenvalues decay super-exponentially for 0<α<3/2

Bounds on eigenvalue counting number for large c

Abstract

In this paper, we first give two uniform asymptotic approximations of the eigenfunctions of the weighted finite Fourier transform operator, defined by ${\displaystyle \mathcal F_c^{(\alpha)} f(x)=\int_{-1}^1 e^{icxy} f(y)\,(1-y^2)^{\alpha}\, dy,\,}$ where $ c >0, \alpha > -1$ are two fixed real numbers. The first uniform approximation is given in terms of a Bessel function, whereas the second one is given in terms of a normalized Jacobi polynomial. These eigenfunctions are called generalized prolate spheroidal wave functions (GPSWFs). By using the uniform asymptotic approximations of the GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the operator $\mathcal F_c^{(\alpha)}$ in the case where $0<\alpha < 3/2.$ Finally, by computing the trace and an estimate of the norm of the operator ${\displaystyle \mathcal Q_c^{\alpha}=\frac{c}{2\pi} \mathcal F_c^{{\alpha}^*} \mathcal F_c^{\alpha},}$ we give a lower and an upper bound for the counting number of the eigenvalues of $Q_c^{\alpha},$ when $c>>1.$