Sampling Theorems for Shift-invariant Spaces, Gabor Frames, and Totally Positive Functions

2016/12/31 by Karlheinz Gröchenig, José Luis Romero, Joachim Stöckler · 80 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Computer vision #Gabor transform #Image and Signal Denoising Methods #Invariant (physics) #Mathematical Analysis and Transform Methods #Mathematics #Pure mathematics #Time–frequency analysis #cs.IT #math.FA #math.IT #msc:42C15 #msc:42C40 #msc:94A20

paper · pdf · open access · doi:10.1007/s00222-017-0760-2

published in Inventiones mathematicae 211(3), 1119-1148 (Springer Science+Business Media) · 25 pages

openalex publication_date 2017/10/20 · arxiv created 2017/10/24 · arxiv updated 2018/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 g(ξ)= ∏_j=1ⁿ (1+2πiδ_jξ)^-1 e^{-c ξ2} for δ₁,…,δ_n∈ 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 αℤ × βℤ forms a frame, if and only if αβ<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.

Citations

• The wavelet transform, time-frequency localization and signal analysis

Cited by

• Gabor frames for rational functions
• Frames for compactly supported functions with irrational density
• Shift-invariant sampling in two-sided small Fock spaces
• Exponential frames and Riesz sequences at the critical density
• Observable sets for Schrödinger equations on combinatorial graphs
• Universal frame set for rational functions
• Frame set for shifted sinc-function
• Time-Frequency Shift Invariance of Gabor Spaces with an S₀-Generator
• Entrywise preservers of sign regularity
• Gaussian Gabor frames, Seshadri constants and generalized Buser--Sarnak invariants
• Frame redundancy and Beurling density
• Conjugate phase retrieval on graphs and with applications in shift-invariant spaces
• Conjugate Phase Retrieval in Paley-Wiener Space
• Constructive Approximation in Mixed norm Spaces
• Localized frames without inequalities
• Multiply positive functions, critical exponent phenomena, and the Jain-Karlin-Schoenberg kernel
• Gabor frames for functions supported on a semi-axis
• Frame sets for generalized B-splines
• Gabor Frames of Totally Positive Functions: A Complete Characterization
• Gabor frames for model sets
• Reconstructing Group Wavelet Transform From Feature Maps With a Reproducing Kernel Iteration. [europepmc]
• Sampling in quasi shift-invariant spaces and Gabor frames generated by ratios of exponential polynomials. [europepmc]

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• Gabor frames for quasi-periodic functions and polyanalytic spaces on the flat cylinder
• Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions
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