Related papers: Shift Invariant Spaces on LCA Groups

We study closed subspaces of $L^2(X)$, where $(X, \mu)$ is a $\sigma$-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group $\Gamma$ on $X$. We provide…

A $(K,\Lambda)$ shift-modulation invariant space is a subspace of $L^2(G)$, that is invariant by translations along elements in $K$ and modulations by elements in $\Lambda$. Here $G$ is a locally compact abelian group, and $K$ and $\Lambda$…

Let G be an LCA group and K a closed subgroup of G. A closed subspace of L^2(G) is called K-invariant if it is invariant under translations by elements of K. Assume now that H is a countable uniform lattice in G and M is any closed subgroup…

For a second countable locally compact group $G$ and a closed abelian subgroup $H$, we give a range function classification of closed subspaces in $L^2(G)$ invariant under left translation by $H$. For a family $\mathscr{A} \subset L^2(G)$,…

Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center (SI/Z), we develop a notion of periodization on the group Fourier transform side, and use this notion to…

In this article we study the structure of $\Gamma$-invariant spaces of $L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is with respect to the action of $\Gamma$, a non commutative group in the form of a semidirect…

The structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group is related to a notion of Hilbert modules endowed with inner products taking values in spaces of unbounded operators. A…

We prove a necessary and sufficient condition for a principal shift invariant space of $L^2(\mathbb{R})$ to be invariant under translations by the subgroup $\frac{1}{N} \mathbb{Z}, N>1$. This condition is given in terms of the Zak transform…

In this article we study invariance properties of shift-invariant spaces in higher dimensions. We state and prove several necessary and sufficient conditions for a shift-invariant space to be invariant under a given closed subgroup of…

The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $\Lambda$-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $\Lambda$ in…

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in $L^2(\R^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior…

In the context of a connected, simply connected, nilpotent Lie group, whose representations are square-integrable modulo the center, we find characterization results of extra-invariant spaces under the left translations associated with the…

In this paper a sampling theory for unitary invariant subspaces associated to locally compact abelian (LCA) groups is deduced. Working in the LCA group context allows to obtain, in a unified way, sampling results valid for a wide range of…

In this paper we study the properties of multiplication invariant (MI) operators acting on subspaces of the vector-valued space $L^2(X;\mathcal H)$. We characterize such operators in terms of range functions by showing that there is an…

Given an $R^d$-valued random field (rf) $Z(t),t\in T$ and an $\alpha$-homogeneous mapping $\kappa$ we define the corresponding equivalent class of rf's (denoted by $K_\alpha$) which include representers of the same tail measure $\nu_Z$.…

We characterize orthonormal bases, Riesz bases and frames which arise from the action of a countable discrete group $\Gamma$ on a single element $\psi$ of a given Hilbert space $\mathcal{H}$. As $\Gamma$ might not be abelian, this is done…

In this article we obtain families of frames for the space B_\omega of functions with band in [-\omega,\omega] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of A. Ron and Z. Shen and a variant,…

Given discrete groups $\Gamma \subset \Delta$ we characterize $(\Gamma,\sigma)$-invariant spaces that are also invariant under $\Delta$. This will be done in terms of subspaces that we define using an appropriate Zak transform and a…

In this note, we describe the backward shift invariant subspaces for a large class of reproducing kernel Hilbert spaces. This class includes in particular de Branges-Rovnyak spaces (the non-extreme case) and the range space of co-analytic…

Successive differences on a sequence of data help to discover some smoothness features of this data. This was one of the main reasons for rewriting the classical interpolation formula in terms of such data differences. The aim of this paper…