Analysis of spectral approximations using prolate spheroidal wave functions

Abstract

Abstract. In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of non-periodic functions in Sobolev spaces. These results serve as an indispensable tool for the analysis of PSWF spectral methods. A PSWF spectral-Galerkin method is proposed and analyzed for elliptic-type equations. Illustrative numerical results consistent with the theoretical analysis are also presented.

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

“…This comes from the fact that Slepian's functions can be written as a series of Legendre polynomials for which fine estimates have been proven on the coefficients [57]. The exact recovery property is shown the same way as in §3, see Theorem 9, and the Corollary 2 is given concerning the recovery of functions belonging to the Hilbert spacesH r c (−1, 1) studied in [55] for which spectral accuracy always holds. Again, numerical tests are displayed in §4.4, involving more complex and possibly noisy signals.…”

“…Thus they can serve as an interpolator on any compact interval of R as an alternative choice which can enjoy spectral accuracy instead of classical polynomial systems like Legendre, see [55] for very precise error estimates in this direction. The following theorem (taken from [57]) summarizes the main properties of (ϕ k ) k∈N as an interpolator: PSWF satisfy also another eigenvalue problem which reads [29,54,57]:…”

“…• approximation of bandlimited functions on R, that is, approximation in P W ω by means of ψ k , k ∈ N which are normalized so as to have ψ k L 2 (R) = 1 (which implies that ϕ k L 2 (−T,T ) = √ λ k → 0 when k grows) as studied for instance in [50], • and approximation in the space L 2 (−T, T ), as presented in Theorem 2 or in [6,57,55], which is made with functions ϕ k normalized so as ϕ k L 2 (−T,T ) = 1 (which implies that…”

“…Following Wang [55], we recall the singular Sturm-Liouville operator associated with the system of PSWF ϕ k , k ∈ N, for a fixed value c > 0 of the Slepian parameter:…”

“…Besides that, these bases allow to perform extrapolation of signals even if this illconditioned problem has to be stabilized (see for instance [17,21]): hence in this perspective, the exact recovery property for sparse signals of CS algorithms may become very valuable as a limited amount of measurements can permit to reconstruct the signal's very disconnected spectrum with supposedly machine's accuracy and then allow to extrapolate observations made in, say, [−1, 1] to a bigger interval (see Remark 8). Usual interpolation properties for PSWF are recalled in §2.2 together with error estimates for spectral approximation [6,15]; in particular, the recent estimates by Wang [55] are included. §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.…”