The Structure of Shift-Invariant Subspaces of L2(Rn)☆

By Marcin Bownik. 2000

288 citations recorded by the source corpus.

Using the range function approach to shift invariant spaces in L2(Rn) we give a simple characterization of frames and Riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2(Zn). This in turn gives a simplified approach to the analysis of frames and Riesz families done by Gramians and dual Gramians. We prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator. As an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/9526f6dc495d4c87fbf236868475299e4fec7163

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Qiyu Sun
2. Akram Aldroubi
3. E. Hern'andez
4. Davide Barbieri
5. Wai-Shing Tang
6. Victoria Paternostro
7. Carlos Cabrelli
8. Q. Jahan

Citations in the corpus

• The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups (2015, unknown): “1 provides a characterization of continuous frames in terms of the fiberization operator by generalizing the corresponding result on Rn by the first author [5] and its LCA extension by Cabrelli and Paternostro [9].”
• Shift Invariant Spaces on LCA Groups (2009, methodology): “develop the theory of shift invariant spaces in LCA groups. Our emphasis will be on range functions and fiberization techniques. The order of the subjects follows mainly the treatment of Bownik in Rd, [1]. In [11] the authors study, in the context of LCA groups, principal shift-invariant spaces, that”
• Subspaces of L2(G) invariant under translation by an abelian subgroup (2014, unknown): “For the subgroup Z ⊆ R, similar results using the so-called fiberization map were given by de Boor, Devore, and Ron [13, 14] and Bownik [6], with wide applications to the theories of approximation, spline systems, wavelets, and Gabor systems.”
• The structure of shift invariant spaces on a locally compact abelian group (2008, unknown): “[1,3,4,6, 7,12,16].”
• Riesz and frame systems generated by unitary actions of discrete groups (2014, methodology): “upp[ψ,ψ]. The characterization of shift invariant spaces in LCA groups was also addressed in [7], in terms of the range function instead of the bracket map. Such tool is strictly related, as shown in [5], to the Gramian analysis of [38, 39]. A characterization in terms of range functions for transla”
• Characterization of wavelets and MRA wavelets on local fields of positive characteristic (2013, methodology): “This characterization was obtained by using the results about shift invariant systems and quasi-affine systems in [8,30,31] and [13].”
• The Zak transform and the structure of spaces invariant by the action of an LCA group (2014, methodology): “This was already known when the group involved acts by translations (see [4, 5, 7]).”
• Convolution, Average Sampling, and a Calderon Resolution of the Identity for Shift-Invariant Spaces (2005, methodology): “To establish the deconvolution formula from Calderon convolution, we need the following result about the generators of a shift-invariant space ([11]).”