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Prolate spheroidal wave functions have recently attracted a lot of attentionin applied harmonic analysis, signal processing and mathematical physics. Theyare eigenvectors of the Sinc-kernel operator Qc : the time-and band-limitingoperator. The corresponding eigenvalues play a key role and it is the aim ofthis paper to obtain precise non-asymptotic estimates for these eigenvalues,within the three main regions of the spectrum of Qc. This issue is rarelystudied in the literature, while the asymptotic behaviour of the spectrum of Qchas been well established from the sixties. As applications of ournon-asymptotic estimates, we first provide estimates for the constantsappearing in Remez and Tur{\`a}n-Nazarov type concentration inequalities. Then,we give an estimate for the hole probability, associated with a random matrixfrom the Gaussian Unitary Ensemble (GUE).

Non-Asymptotic behaviour of the spectrum of the Sinc Kernel Operator and Related Applications