Weighted Finite Fourier Transform Operator: Uniform Approximations of the Eigenfunctions, Eigenvalues Decay and Behaviour

In this paper, we investigate two uniform asymptotic approximations as well as some spectral properties of the eigenfunctions of the weighted finite Fourier transform operator, defined by Here, are two fixed real numbers. These eigenfunctions are called generalized prolate spheroidal wave functions (GPSWFs) and they are firstly introduced and studied in Wang and Zhang (Appl Comput Harmon Anal 29(3):303-329, 2010). The present study is motivated by the promising concrete applications of the GPSWFs in various scientific area such as numerical analysis, mathematical physics and signal processing. We should mention that these two uniform approximation results of the GPSWFs can be considered as generalizations of the results given in the joint work of one of us (Bonami and Karoui in Constr Approx 43(1):15-45, 2016). As it will be seen, these generalizations require some involved extra work, especially in the case where By using the uniform asymptotic approximations of the GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the operator in the case where Moreover, by computing the trace and an estimate of the norm of the operator we give a lower bound for the counting number of the eigenvalues of when Finally, we provide the reader with some numerical examples that illustrate the different results of this work.