Slanted matrices, Banach frames, and sampling

In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in $\ell^p$ for some $p$ implies boundedness below in $\ell^p$ for all $p$. We use the established resultto enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matrices

Linear combinations of generators in multiplicatively invariant spaces

Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega,\mathcal{H})$ that are invariant under multiplications of (some) functions in $L^{\infty}(\Omega)$. In this paper we work with MI spaces that are finitely generated. We prove that almost every linear combination of the generators of a finitely generated MI space produces a new set on generators for the same space and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply what we prove for MI spaces to system of translates in the context of locally compact abelian groups and we obtain results that extend those previously proven for systems of integer translates in $L^2(\mathbb{R}^d)$.Comment: 13 pages. Minor changes have been made. To appear in Studia Mathematic

Riesz bases in L2(0,1) related to sampling in shift-invariant spaces

AbstractThe Fourier duality is an elegant technique to obtain sampling formulas in Paley–Wiener spaces. In this paper it is proved that there exists an analogue of the Fourier duality technique in the setting of shift-invariant spaces. In fact, any shift-invariant space Vφ with a stable generator φ is the range space of a bounded one-to-one linear operator T between L2(0,1) and L2(R). Thus, regular and irregular sampling formulas in Vφ are obtained by transforming, via T, expansions in L2(0,1) with respect to some appropriate Riesz bases

On Some Sampling-Related Frames in U-Invariant Spaces

This paper is concerned with the characterization as frames of some sequences in -invariant spaces of a separable Hilbert space H where U denotes an unitary operator defined on H ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces in L2(R) , where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operator to belong to a continuous group of unitary operators

Perturbed sampling formulas and local reconstruction in shift invariant spaces

AbstractLet Vϕ be a closed subspace of L2(R) generated from the integer shifts of a continuous function ϕ with a certain decay at infinity and a non-vanishing property for the function Φ†(γ)=∑n∈Zϕ(n)e−inγ on [−π,π]. In this paper we determine a positive number δϕ so that the set {n+δn}n∈Z is a set of stable sampling for the space Vϕ for any selection of the elements δn within the ranges ±δϕ. We demonstrate the resulting sampling formula (called perturbation formula) for functions f∈Vϕ and also we establish a finite reconstruction formula approximating f on bounded intervals. We compute the corresponding error and we provide estimates for the jitter error as well