The construction of wavelets from generalized conjugate mirror filters in L2(Rn)

Baggett, Larry; Courter, Jennifer E.; Merrill, Kathy D.

doi:10.1016/s1063-5203(02)00509-2

Cited by 27 publications

(53 citation statements)

References 11 publications

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“…which is a degenerate version of the original multiplicity function m. Thus one does not have an analogue of Theorem 3.4 of [2] in this situation, and one must rely on the direct limit Hilbert space instead of L 2 (R) to construct our GMRA. In all of these cases, the infinite product ∞ j=1 [2 −1/2 H(2 −j x)] gives functions whose inverse Fourier transforms in L 2 (R) generate a shift invariant subspace with multiplicity function a degenerate form of the original m.…”

Section: Examplesmentioning

confidence: 99%

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: Examplesmentioning

confidence: 99%

Generalized Multiresolution Analyses with Given Multiplicity Functions

Baggett

Larsen

Merrill

et al. 2008

J Fourier Anal Appl

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“…The geometry of orthonormal wavelets is fairly well understood [30,15,18,19,26,29]. The main technique used in the study of orthonormal wavelets is the multiresolution analysis (MRA) introduced by Mallat and Meyer [3,8] and its generalizations [1,4,5].…”

Section: Introductionmentioning

confidence: 99%

“…Any representation of the Baumslag-Solitar group B S (1,2) is in fact given by two unitary operators U and T on some Hilbert space H, that satisfy the relation U T U −1 = T 2 . While the Baumslag-Solitar group appears to be quite simple, this can be deceiving, there are several extremely interesting results about it in the literature which reveal surprising properties.…”

Section: Introductionmentioning

confidence: 99%

“…A wavelet set is a wavelet ψ such that its Fourier transformψ is a characteristic function. In [14,17] many examples of Parseval wavelet sets are provided where the orthonormal dilation lies in the space L 2 (R) ⊕· · · ⊕ L 2 (R) with the representation of the group B S (1,2) given by a simple amplification of the representation {U 0 , T 0 } in L 2 (R):Ũ = U 0 ⊕ · · · ⊕ U 0 , T = T 0 ⊕ · · · ⊕ T 0 . There is one issue with this representation, as shown in [17]: it does not have orthogonal multiresolution wavelets.…”

Section: Introductionmentioning

confidence: 99%

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Orthonormal dilations of Parseval wavelets

et al. 2008

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We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar groupWe give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics (Theorem 3.14). We prove that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics (Theorem 3.24). We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we construct Parseval wavelet sets which have infinitely many non-isomorphic orthonormal dilations.

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“…Our paper focuses on R W in the wavelet context; and we demonstrate that for many wavelets, the weight function must take the form of an operator transformation X → mXm * where m is a fixed matrix valued function and where m * denotes the adjoint operator, in this case transpose-conjugate. This matrix version of R W is necessary for understanding wavelet constructions associated to wavelet sets [15], and more generally to non-MRA wavelets, [3]. (MRA stands for multiresolution analysis [16].…”

Section: Introductionmentioning

confidence: 99%

Covariant Representations for Matrix-valued Transfer Operators

Dutkay

Røysland

2008

Integr. equ. oper. theory

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Motivated by the multivariate wavelet theory, and by the spectral theory of transfer operators, we construct an abstract affine structure and a multiresolution associated to a matrix-valued weight. We describe the one-toone correspondence between the commutant of this structure and the fixed points of the transfer operator. We show how the covariant representation can be realized on R n if the weight satisfies some low-pass condition.

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The construction of wavelets from generalized conjugate mirror filters in L2(Rn)

Cited by 27 publications

References 11 publications

Generalized Multiresolution Analyses with Given Multiplicity Functions

Generalized Multiresolution Analyses with Given Multiplicity Functions

Orthonormal dilations of Parseval wavelets

Covariant Representations for Matrix-valued Transfer Operators

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