Spectral decay of time and frequency limiting operator

For fixed c, the Prolate Spheroidal Wave Functions (PSWFs) ψn,c form a basis with remarkable properties for the space of band-limited functions with bandwidth c. They have been largely studied and used after the seminal work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications rely heavily on the behavior and the decay rate of the eigenvalues (λn(c))n≥0 of the time and frequency limiting operator, which we denote by Qc. Hence, the issue of the accurate estimation of the spectrum of this operator has attracted a considerable interest, both in numerical and theoretical studies. In this work, we give an explicit integral approximation formula for these eigenvalues. This approximation holds true starting from the plunge region where the spectrum of Qc starts to have a fast decay. As a consequence of our explicit approximation formula, we give a precise description of the super-exponential decay rate of the λn(c). Also, we mention that the described approximation scheme provides us with fairly accurate approximations of the λn(c) with low computational load, even for very large values of the parameters c and n. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.