Abstract
We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.
Highlights
1.1 Nonuniform samplingBeurling’s sampling theory for the Paley–Wiener space is at the crossroad of complex analysis and signal processing and has served as a model and inspiration for many generations of sampling theorems in both fields
Austria subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice αZ × βZ forms a frame, if and only if αβ < 1
The duality theory of Gabor frames [51] in the formulation of Janssen [35] relates the frame property of G(g, αZ × βZ) to a sampling problem in the shift-invariant space V 2(g). We extend this connection to nonuniform sampling sets and derive a characterization of semi-regular Gabor frames, which we formulate in a coarse version as follows
Summary
Beurling’s sampling theory for the Paley–Wiener space is at the crossroad of complex analysis and signal processing and has served as a model and inspiration for many generations of sampling theorems in both fields. It is expected that nonuniform sampling theorems similar to Theorem 1.1 hold for general shift-invariant spaces Such claims are usually backed by a heuristic comparison to the bandlimited case, but, in spite of their central role in signal processing, they have been given only moderate formal support so far. Theorem 1.2 validates the heuristic understanding of the signal processing community that nonuniform sampling above the critical density (the Nyquist rate) leads to stable reconstruction in a shift-invariant space. We stress that Theorem 1.2 is new even for the shift-invariant space with Gaussian generator We believe that it opens a new avenue in approximation theory of radial basis functions, because sampling inequalities have become an integral part of the error analysis for scattered data interpolation [30,48].
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