Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ?n

Baggett, Larry; Medina, Herbert A.; Merrill, Kathy D.

doi:10.1007/bf01257191

Cited by 139 publications

(153 citation statements)

References 8 publications

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“…GMRAs were introduced in [4], and have since been studied in [5] and [3]; other authors had previously observed that much of the theory still works when we use Γ in place of the classical translation group Z n (see [1] and [9], for example).…”

Section: Multiplicity Functions and The Main Theoremmentioning

confidence: 99%

“…Ak and π is the representation determined by translation, the measure µ is necessarily absolutely continuous with respect to the Haar measure on the torus [4,Propositions 2 and 3]). This absolute continuity does not necessarily hold in general, but here we are interested in the converse, and we assume that our measures µ are absolutely continuous with respect to the Haar measure λ on Γ.…”

Section: Multiplicity Functions and The Main Theoremmentioning

confidence: 99%

“…Generalized multiresolution analyses were introduced by Baggett, Medina and Merrill in [4], where they were used to construct many examples of wavelet sets, including the one whose characteristic function is the Fourier transform of the Journè wavelet. A generalized multiresolution analysis (GMRA) for a Hilbert space H consists of an increasing sequence of closed subspaces V n such that the complements W n := V n+1 V n give a directsum decomposition H = n∈Z W n in which the W n for n ≥ 0 are invariant under a representation π of an abelian group Γ on H, and in which W n+1 is the dilation of W n .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized Multiresolution Analyses with Given Multiplicity Functions

Baggett

Larsen

Merrill

et al. 2008

J Fourier Anal Appl

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Abstract. Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L 2 (R n ), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.

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Section: Multiplicity Functions and The Main Theoremmentioning

confidence: 99%

Section: Multiplicity Functions and The Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Multiresolution Analyses with Given Multiplicity Functions

Baggett

Larsen

Merrill

et al. 2008

J Fourier Anal Appl

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“…Before proceeding, we remark that the analysis of wavelets using the spectral multiplicity methods was begun by Baggett, Medina, and Merrill [2] in their study of generalized multiresolution analyses and MSF wavelets. Other spectral methods in the analysis of wavelets can be found in Jorgensen, et al in [7,20].…”

Section: Affine Quasi-affine and Weyl Heisenberg Framesmentioning

confidence: 99%

Geometric aspects of frame representations of abelian groups

Aldroubi

Larson

Tang

et al. 2004

Trans. Amer. Math. Soc.

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Abstract. We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

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“…The multiplicity function is presented there as a complete invariant. Of course, in wavelet applications, there are also the additional consistency relations (see [BMM99]), but the notion of a multiplicity function is general.…”

Section: Unitary Operatorsmentioning

confidence: 99%

Optimal Decompositions of Translations of L 2-Functions

Jørgensen

Song

2008

Complex anal.oper. theory

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Abstract. In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space L 2 (R n ). Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space H, or equivalent the spectral theory of a unitary representation U of the rank-n lattice Z n in R n . Starting with a non-zero vector ψ ∈ H, we look for relations among the vectors in the cyclic subspace in H generated by ψ. Since these vectors {U (k)ψ|k ∈ Z n } involve infinite "linear combinations," the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name L 2 -independence. This refers to infinite linear combinations of integral translates of a fixed function with l 2 -coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals.Mathematics Subject Classification (2000). Primary 47B40, 47B06, 06D22, 62M15; Secondary 42C40, 62M20.

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Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ?n

Cited by 139 publications

References 8 publications

Generalized Multiresolution Analyses with Given Multiplicity Functions

Generalized Multiresolution Analyses with Given Multiplicity Functions

Geometric aspects of frame representations of abelian groups

Optimal Decompositions of Translations of L 2-Functions

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