The structure of shift–modulation invariant spaces: The rational case

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.

Optimal shift invariant spaces and their Parseval frame generators

In this paper, a shift-invariant system of the form $$\{L_{a\nu }g_s:\nu \in \varGamma ,s\in {{\mathbb {Z}}}\}$$ for $$a>0$$ is studied on the Heisenberg group $${{\mathbb {H}}^n}$$ , where $$L_x$$ denotes the left translation operator on $${{\mathbb {H}}^n}$$ and $$\varGamma $$ is a lattice in $${{\mathbb {H}}^n}$$ . The characterizations for the shift-invariant system to be a Bessel sequence and a frame sequence are given in terms of certain operators arising from the fiber map corresponding to this system. For a shift invariant system to be a frame sequence (Riesz sequence) the characterization is given in terms of dual Gramian (Gramian). Moreover, the problem of characterizing a pair of shift-invariant systems to be dual frames is also discussed.

Backward shift operator and finite dimensional de Branges Rovnyak spaces in the ball

Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space \(V_q(\Phi, \Lambda)\) modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wide-band communication. In particular, the space \(V_q(\Phi, \Lambda)\) is generated by a family of well-localized molecules \(\Phi\) of similar size located on a relatively separated set \(\Lambda\) using \(\ell^q\) coefficients, and hence is locally finitely generated. Moreover that space \(V_q(\Phi, \Lambda)\) includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator \(\Phi\), we show that if the generator \(\Phi\) is a frame for the space \(V_2(\Phi, \Lambda)\) and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator \(\Phi\) for the space \(V_q(\Phi, \Lambda)\) with \(q\ne 2\), and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.

Recovery of missing samples for oversampling in shift invariant spaces

The aim of this paper is to study sampling and reconstruction in a shift invariant space with multiple generators. On contrary to the classical case of a shift invariant space with a single generator, it is shown that cannot be a stable set of sampling for where when Further the problems of perturbation of a stable set of sampling and local reconstruction method are discussed along with an illustration and implementation.

Extra invariance of shift-invariant spaces on LCA groups

We define the output-nulling (ON) (A,E,B)-invariant subspaces for singular systems, showing their dynamic significance. A Generalized Lyapunov equation is introduced to compute such a subspace, the feedback required to assign it as a closed-loop invariant subspace, and also the spectrum assigned by that feedback on the subspace. A definition of teachability subspace for singular systems is given. A new singular system structure algorithm is related to the supremal ON (A,E,B)- invariant subspace.

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Publication in the conference proceedings of EUSIPCO, Marrakech, Morocco, 2013

Generalized sampling in shift-invariant spaces with multiple stable generators

It is the purpose of this paper to feature the link between the theory of minimal norm interpolation over lattices by elements fromSobolev algebras H s (S d )with what is known as the theory of spline-type (or principal shift invariant) spaces. As an extremely useful tool allowing us to establish various kinds of robustness results we will present the Wiener amalgam spacesW(B, l p with general (smooth) local components and globall p behavior. For this reason a summary of their most important properties, including convolution relations and the behavior under the Fourier transform, is presented. The discussion of projection and minimal norm interpolation operators is not restricted to the pure Hilbert space setting for which these concepts were developed originally. Among others we show LP-stability of the (for p = 2 orthogonal) projection from LP onto the corresponding spline-type spaces with lP-coefficients. As a main result (which can be formulated in several different concrete ways) we show that for s > d/2 the mapping f ↦ Qs, a(f), from f to the minimal norm interpolation in Hs over the lattice aℤd, a >, depends continuously on the input parameters (s, a). It also extends to certain fractional LP-Sobolev spaces consisting of continuous functions in LP. In this modified setting the outcome of this procedure depends continuously on (s, a) in the LP sense. Moreover, the mapping is robust against small jitter errors. Wiener amalgam spaces tum out to be very useful, both for a precise formulation and in the proofs of such results.

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for frames generated by any compactly supported function φ whose integer-translates form a partition of unity, e.g., a B-spline, we construct a class of dual frame generators, formed by linear combinations of translates of φ. This allows us to chose a dual generator with special properties, for example, the one with shortest support, or a symmetric one in case the frame itself is generated by a symmetric function. One of these dual generators has the property of being constant on the support of the frame generator.

Approximation from shift-invariant spaces by generalized sampling formulas