The infimum cosine angle between two finitely generated shift-invariant spaces and its applications

Kim, Hong Oh; Kim, Rae Young; Lim, Jae Kun

doi:10.1016/j.acha.2005.05.002

Cited by 16 publications

(14 citation statements)

References 28 publications

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“…One of the advantage of considering duals outside the space M is that, in some situations, it allows the dual functions to have better localization properties, both in the time and frequency domain, then those that belong to the original space. (See [12,13,16] for motivation and results in that direction.) Those two kinds of requirements lead to the definition of shift-generated duals of type I and type II, respectively.…”

Section: Introductionmentioning

confidence: 99%

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Hemmat

Gabardo

2007

J Fourier Anal Appl

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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.

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Section: Introductionmentioning

confidence: 99%

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Hemmat

Gabardo

2007

J Fourier Anal Appl

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“…Proof of Theorem 1.6 Assuming the hypotheses of Theorem 1.6, it follows from [19,Lem. 4.9] and its proof that 2 (Z d ) = U ||ξ ( V ||ξ ) ⊥ for a.e.…”

Section: Proposition 73mentioning

confidence: 98%

Duals of Frame Sequences

2008

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Frames provide unconditional basis-like, but generally nonunique, representations of vectors in a Hilbert space H . The redundancy of frame expansions allows the flexibility of choosing different dual sequences to employ in frame representations. In particular, oblique duals, Type I duals, and Type II duals have been introduced in the literature because of the special properties that they possess. A Type I dual is a dual such that the range of its synthesis operator is contained in the range of the synthesis operator of the original frame sequence, and a Type II dual is a dual such that the range of its analysis operator is contained in the range of the analysis operator of the original frame sequence. This paper proves that all Type I and Type II duals are oblique duals, but not conversely, and characterizes the existence of oblique and Type II duals in terms of direct sum decompositions of H , as well as characterizing when the Type I, Type II, and oblique duals will be unique. These results are also applied to the case of shift-generated sequences that are frames for shift-invariant subspaces of L 2 (R d ).

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“…See [10] for an application of these angles to the perturbation of frame sequences. The infimum cosine angle is closely related with the bi-orthogonality of two multiresolution analyses (MRAs) [1,2,7,[21][22][23][29][30][31]. In particular, the authors have recently found a useful expression of the infimum cosine angle between finitely generated shiftinvariant subspaces in terms of the Gramians of generating sets in a companion paper [23].…”

Section: Introductionmentioning

confidence: 99%

“…The infimum cosine angle is closely related with the bi-orthogonality of two multiresolution analyses (MRAs) [1,2,7,[21][22][23][29][30][31]. In particular, the authors have recently found a useful expression of the infimum cosine angle between finitely generated shiftinvariant subspaces in terms of the Gramians of generating sets in a companion paper [23]. On the other hand, the supremum cosine angle is closely related with the closedness of the sum of two closed subspaces of a Hilbert space, as can be seen in the following proposition by Tang.…”

Section: Introductionmentioning

confidence: 99%

Characterization of the closedness of the sum of two shift-invariant spaces

Kim

Lim

2006

Journal of Mathematical Analysis and Applications

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We first present a formula for the supremum cosine angle between two closed subspaces of a separable Hilbert space under the assumption that the 'generators' form frames for the subspaces. We then characterize the conditions that the sum of two, not necessarily finitely generated, shiftinvariant subspaces of L 2 (R d ) be closed. If the fibers of the generating sets of the shift-invariant subspaces form frames for the fiber spaces a.e., which is satisfied if the shift-invariant subspaces are finitely generated or if the shifts of the generating sets form frames for the respective subspaces, then the characterization is given in terms of the norms of possibly infinite matrices. In particular, if the shift-invariant subspaces are finitely generated, then the characterization is given wholly in terms of the norms of finite matrices.

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The infimum cosine angle between two finitely generated shift-invariant spaces and its applications

Cited by 16 publications

References 28 publications

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Duals of Frame Sequences

Characterization of the closedness of the sum of two shift-invariant spaces

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