Eigenvalue distribution of time and frequency limiting

Time-Frequency Localization and Sampling of Multiband Signals

“…Landau and Widom [24] introduced a method to estimate the distribution of eigenvalues for P Σ Q S asymptotically in the sense of rescaling one of the sets S or Σ when both sets are finite unions of intervals. Their result, Theorem 3, confirmed a conjecture of D. Slepian [36] asserting that the number of eigenvalues of P Ω Q T is approximately c = ΩT when c is large.…”

“…Fourier uniqueness now implies that the functions ϕn are complete in In their works [29,34,24], Landau, Slepian, Pollak and Widom proved a number of statements of the "ΩT " stating, in essence, that the dimension of the space of essentially T -timelimited and Ω-bandlimited signals is essentially the time-frequency area ΩT . One version-Theorem 3 below-says that P Ω Q T has about ΩT eigenvalues close to one, and that the eigenvalues plunge rapidly from λ ≈ 1 to λ ≈ 0 over a transition band of width around log ΩT .…”

“…The individual terms in this sum are O(1) (uniformly bounded independent of c) unless i = k and Γ k is adjacent to ∆ j (see the Lemma in [24]). Consequently one can express…”

“…We begin reviewing the work of Landau, Slepian and Pollak [35,28,29,32] -the so-called Bell Labs theory -concerning "time-and bandlimiting" (see also [22,24,34,26]) in Section 2. The Paley-Wiener space PW consists of those functions in L 2 (R) whose Fourier transforms vanish outside the set [−1/2, 1/2].…”

Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions

“…§3 is devoted to proving a L ∞ bound on a subset of the PSWF base on [−1, 1]; like Legendre polynomials, PSWF can display sharp "tails" close to the edges of this interval. However, the situation here is better compared to polynomials because there exists a collection of indexes for which both a L ∞ bound and spectral accuracy hold as stated in Lemma 1; roughly speaking, it corresponds to the PSWF endowed with eigenvalues not too far from 1 (this statement can be made precise by means of the classical Landau-Widom estimate, see [32]). With this L ∞ bound at hand, it is possible to follow the canvas of [11] and estimate the concentration measure parameter µ which leads to the RIP under technical assumptions.…”