Non-asymptotic behavior of the spectrum of the sinc-kernel operator and related applications

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 Qc: 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 Qc. This issue is rarely studied in the literature, while the asymptotic behavior of the spectrum of Qc 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 Turan–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.

[1]  A. Bonami,et al.  Approximations in Sobolev Spaces by Prolate Spheroidal Wave Functions , 2015, 1509.02651.

[2]  D. Slepian Some Asymptotic Expansions for Prolate Spheroidal Wave Functions , 1965 .

[3]  Andrei Osipov Certain upper bounds on the eigenvalues associated with prolate spheroidal wave functions , 2012 .

[4]  Laurent Gosse,et al.  Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions , 2013 .

[5]  D. Slepian Some comments on Fourier analysis, uncertainty and modeling , 1983 .

[6]  John P. Boyd,et al.  Approximation of an analytic function on a finite real interval by a bandlimited function and conjectures on properties of prolate spheroidal functions , 2003 .

[7]  H. Pollak,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals , 1962 .

[8]  H. Widom Asymptotic behavior of the eigenvalues of certain integral equations. II , 1964 .

[9]  T. Tao,et al.  Random matrices: The Universality phenomenon for Wigner ensembles , 2012, 1202.0068.

[10]  M. Mehta,et al.  Asymptotic behavior of spacing distributions for the eigenvalues of random matrices , 1973 .

[11]  Abderrazek Karoui,et al.  Spectral Decay of Time and Frequency Limiting Operator , 2015 .

[12]  H. Hedenmalm,et al.  The Polyanalytic Ginibre Ensembles , 2011, 1106.2975.

[13]  Li-Lian Wang,et al.  A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods , 2017 .

[14]  Jochen Trumpf,et al.  Degrees of Freedom of a Communication Channel: Using DOF Singular Values , 2010, IEEE Transactions on Information Theory.

[15]  C. Tracy,et al.  Introduction to Random Matrices , 1992, hep-th/9210073.

[16]  A. Bonami,et al.  Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions , 2014, 1405.3676.

[17]  H. Widom The Asymptotics of a Continuous Analogue of Orthogonal Polynomials , 1994 .

[18]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[19]  F. Dyson Statistical Theory of the Energy Levels of Complex Systems. I , 1962 .

[20]  H. Landau The eigenvalue behavior of certain convolution equations , 1965 .

[21]  Freeman J. Dyson,et al.  Fredholm determinants and inverse scattering problems , 1976 .

[22]  Alexander Its,et al.  A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics , 1997 .

[23]  J. Boyd Prolate spheroidal wavefunctions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudospectral algorithms , 2004 .

[24]  Felix Krahmer,et al.  Spectral Methods for Passive Imaging: Non-asymptotic Performance and Robustness , 2017, SIAM J. Imaging Sci..

[25]  Abderrazek Karoui,et al.  The approximation of almost time- and band-limited functions by their expansion in some orthogonal polynomials bases , 2015, J. Approx. Theory.

[26]  Li-Lian Wang,et al.  Analysis of spectral approximations using prolate spheroidal wave functions , 2009, Math. Comput..

[27]  H. Landau On the density of phase-space expansions , 1993, IEEE Trans. Inf. Theory.

[28]  W. Fuchs On the eigenvalues of an integral equation arising in the theory of band-limited signals , 1964 .

[29]  H. Landau,et al.  Eigenvalue distribution of time and frequency limiting , 1980 .

[30]  Asymptotics of level-spacing distributions for random matrices. , 1992, Physical review letters.