José Luis Romero

Deformation of Gabor systems

Abstract We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurlings characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames “without inequalities” from lattices to non-uniform sets.

Multi-window Gabor frames in amalgam spaces

We show that multi-window Gabor frames with windows in the Wiener algebra

Density of sampling and interpolation in reproducing kernel Hilbert spaces

W(L^{\infty}, \ell^{1})

Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions

are Banach frames for all Wiener amalgam spaces. As a byproduct of our results we positively answer an open question that was posed by [Krishtal and Okoudjou, Invertibility of the Gabor frame operator on the Wiener amalgam space, J. Approx. Theory, 153(2), 2008] and concerns the continuity of the canonical dual of a Gabor frame with a continuous generator in the Wiener algebra. The proofs are based on a recent version of Wieners

The Weyl–Heisenberg ensemble: hyperuniformity and higher Landau levels

1/f

Stability of Gabor Frames Under Small Time Hamiltonian Evolutions

lemma.

Completeness of Gabor systems

We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is small compared to the volume of balls (weak annular decay property) and if the kernel possesses some off-diagonal decay or even some weaker form of localization, then there exists a critical density D with the following property: a set of sampling has density ⩾D, whereas a set of interpolation has density ⩽D. The main theorem unifies many known density theorems in signal processing, complex analysis, and harmonic analysis. For the special case of bandlimited function we recover Landaus fundamental density result. In complex analysis we rederive a critical density for generalized Fock spaces. In harmonic analysis we obtain the first general result about the density of coherent frames.

The Weyl-Heisenberg ensemble: Statistical mechanics meets time-frequency analysis

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

Multitaper spectral estimation and off-grid compressive sensing: MSE estimates

\hat{g}(\xi )= \prod _{j=1}^n (1+2\pi i\delta _j\xi )^{-1} \, e^{-c \xi ^2}