Journal of Mathematical Physics, Volume 62, Issue 3, March 2021. Prolate spheroidal wave functions have recently attracted much attention in applied harmonic analysis, signal processing, and mathematical physics. They are eigenvectors of the sinc-kernel operator [math]: the time- and band-limiting operator. The corresponding eigenvalues play a key role, and the aim of this paper is to obtain precise non-asymptotic estimates with explicit constants related to the spectrum of [math]. This issue is rarely studied in the literature, while the asymptotic behavior of the spectrum of [math] has been well established from the 1960s. However, many recent applications require such non-asymptotic behavior. As applications of our non-asymptotic estimates, we first provide estimates for the constants appearing in the Remez- and Turàn–Nazarov-type concentration inequalities. Then, we give a non-asymptotic upper bound for the gap probability of the sinc determinantal point process. Consequently, one gets a non-asymptotic estimate for the hole probability, associated with bulk scaled asymptotics of a random matrix from the Gaussian unitary ensemble. This last result can be considered as a complement of the various and more involved asymptotic counterparts of this estimate.

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