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 ${\mathcal{F}}_c^{(\alpha )}f(x)={\int }_{-1}^1{e}^{icxy}f(y)(1-y^2{)}^{\alpha }dy,$ 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 ${\mathcal{Q}}_c^{\alpha }=\frac{c}{2\pi }{\mathcal{F}}_c^{{\alpha }^{\ast }}{\mathcal{F}}_c^{\alpha },$ we give a lower and an upper bound for the counting number of the eigenvalues of $Q_c^{\alpha },$ when $c>>1.$

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• Spherical Slepian functions and the polar gap in geodesy
• Spectral Decay of Time and Frequency Limiting Operator
• Uniform approximation and explicit estimates for the prolate spheroidal wave functions
• Generalized Prolate Spheroidal Wave Functions: Spectral Analysis and Approximation of Almost Band-limited Functions