Given an invertible n × n matrix B and a finite or countable subset of L 2 (R n ), we consider the collection X = {φ(· − Bk) : φ ∈ , k ∈ Z n } generating the closed subspace M of L 2 (R n ). If that collection forms a frame for M, one can introduce two different types of shift-generated (SG) dual frames for X, called type I and type II SG-duals, respectively. The main distinction between them is that a SG-dual of type I is required to be contained in the space M generated by the original frame while, for a type II SG-dual, one imposes that the range of the frame transform associated with the dual be contained in the range of the frame transform associated with the original frame. We characterize the uniqueness of both types of duals using the Gramian and dual Gramian operators which were introduced in an article by Ron and Shen and are known to play an important role in the theory of shift-invariant spaces.
Abstract. Let h be a generalized frame in a separable Hilbert space H indexed by a measure space (M, S, µ), and assume its analysing operator is surjective. It is shown that h is essentially discrete; that is, the corresponding index measure space (M, S, µ) can be decomposed into atoms E 1 , E 2 , · · · such that L 2 (µ) is isometrically isomorphic to the weighted space 2 w of all sequences {c i } of complex numbers with ||{c i }|| 2 = |c i | 2 w i < ∞, whereThis provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in L 2 (R).
Gramian analysis is used to study properties of a shift-invariant system X = {φ(· − Bk): φ ∈ Φ, k ∈ Z n }, where B is an invertible n × n matrix and Φ a finite or countable subset of L 2 (R n ) under the assumption that the system forms a frame for the closed subspace M of L 2 (R n ). In particular, the relationship between various features of such system, such as being a frame for the whole space L 2 (R n ), being a Riesz sequence and having a unique shift-generated dual of type I or II is discussed in details. Several interesting examples are presented.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.