The Structure of Finitely Generated Shift-Invariant Spaces in L2(IR(d))

Abstract

A simple characterization is given of finitely generated subspaces of L 2 (IRd) which are invariant under translation by any (multi)integer, and used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence. The last property makes sense only for 'local' spaces, i.e., shift-invariant spaces generated by finitely many compactly supported functions, and special attention is paid to such spaces.As an application, we prove that the approximation order provided by a given local space is already provided by the shift-invariant space generated by just one function, with this function constructible as a finite linear combination of the finite generating set for the whole space, hence compactly supported. This settles a question of some 20 years' standing.

Stable Filtering Schemes with Rational Dilations

“…This innocent consequence of refinability does not hold, in general, for non-integer dilation factors and presents perhaps the most challenging obstacle in generalizing MRA constructions to include rational dilation factors. This can be seen in the work of Auscher [1] as well as the exposition of MRAs for the dilation a = 3 2 offered by Daubechies in [6]. Furthermore, in [1], Auscher proved:…”

“…The theory of shift-invariant spaces has been extensively described in the literature, e.g., [2,3,4,11], yet it will be convenient to establish a minimal amount of machinery in order to naturally develop the results of subsequent sections.…”

Time Frequency Representations of Almost Periodic Functions

“…We wanted to understand whether this is an artifact of the specific approaches that were chosen in those articles or, perhaps, there is a deeper connection between L 2 -representations and AP-representations. To this end, we investigated the problem using the fiberization tools that were developed in the context of general shift-invariant systems, [BDR94], [RS95], and the specific results that followed for WH systems, [RS97a] and wavelet systems, [RS97b].…”

Operators commuting with a discrete subgroup of translations

“…The closed span of G in some L p is a shift-invariant space. For shift-invariant systems, see [2,4,18,30]. On a theoretical level, the main objectives are to understand the spanning and stability properties of G. These are encoded in the spectrum of the frame operator associated to G. More generally, given the shift-invariant systems G and H = {T ak h j : k ∈ Z n , j ∈ I }, we define the frame type operator S = S G,H by…”

“…If ϕ ∈ S(R n ), it is easy to see that the sequence c k = ϕ, T ak g belongs to 2 (Z n ). If I is finite, then Sϕ, ψ = k∈Z n j ∈I ϕ, T ak g j ψ, T ak h j converges absolutely for ϕ, ψ ∈ S(R n ) and, as a consequence, S is continuous from S(R n ) into S (R n ).…”