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

Abstract

Prolate spheroidal wave functions have recently attracted a lot of attention in applied harmonic analysis, signal processing and mathematical physics. They are eigenvectors of the Sinc-kernel operator Qc : the time-and band-limiting operator. The corresponding eigenvalues play a key role and it is the aim of this paper to obtain precise non-asymptotic estimates for these eigenvalues, within the three main regions of the spectrum of Qc. This issue is rarely studied in the literature, while the asymptotic behaviour of the spectrum of Qc has been well established from the sixties. As applications of our non-asymptotic estimates, we first provide estimates for the constants appearing in Remez and Tur{à}n-Nazarov type concentration inequalities. Then, we give an estimate for the hole probability, associated with a random matrix from the Gaussian Unitary Ensemble (GUE).

References in corpus (5)

• The polyanalytic Ginibre ensembles
• Spectral Decay of Time and Frequency Limiting Operator
• The approximation of almost time and band limited functions by their expansion in some orthogonal polynomials bases
• Uniform approximation and explicit estimates for the prolate spheroidal wave functions
• Approximations in Sobolev Spaces by Prolate Spheroidal Wave Functions

Cited by in corpus (4)

• Quantum Fourier Transform Has Small Entanglement
• Increasing resolution and instability for linear inverse scattering problems
• Entanglement entropies of an interval in the free Schrödinger field theory on the half line
• Entanglement entropies of an interval in the free Schrödinger field theory at finite density