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

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 $c>>1.$