Compactly Supported Wavelets and Representations of the Cuntz Relations

Bratteli, Ola; Evans, David; Jørgensen, Palle E. T.

doi:10.1006/acha.2000.0283

Cited by 35 publications

(50 citation statements)

References 25 publications

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“…Extensive analysis has been conducted on these and other related wavelet representations (see [3,4,15,16] for instance). When the scaling function ϕ is compactly supported, the associated representation possesses a finite-dimensional subspace K which satisfies (i) and (ii) above for the isometries S i .…”

Section: Applications To Representation Theory For O Nmentioning

confidence: 99%

QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

Kribs¹

2003

Proceedings of the Edinburgh Mathematical Society

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We show that the representations of the Cuntz C * -algebras On which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finitedimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work. There has been considerable recent interest in the analysis of completely positive maps on finite-dimensional space. There are a number of reasons for this including connections with wavelet analysis [3,4,15], dilation theory [11,16], representation theory of the Cuntz C * -algebras O n [3,4,10,11], and quantum information theory [1,17,21,22]. The results obtained in the current paper have implications for each of these areas. In presenting this work, another goal we have is to push further the connections between the various areas mentioned above.In § 1 we establish a result for completely positive maps. While we focus on the finitedimensional setting, this is not necessary in the proof. A structure theorem for the fixedpoint set of a unital quantum channel is contained in § 2. In particular, we prove that the fixed-point set is a C * -algebra which is equal to the commutant of the algebra generated by any choice of row contraction which determines the channel. We discuss the twodimensional channels [17,21], and use the theorem to classify them by their fixed-point sets. The representation theory for O n is considered in § 3. We focus on a subclass of representations arising in dilation theory and wavelet analysis [3, 4, 10, 11, 15]. Each of these representations determines a completely positive map on finite-dimensional space. We ask if these representations can be classified just by examining the map. An affirmative answer is provided by the result on completely positive maps from the first section. Finally, in § 4 we pose some open questions which are motivated by this work.

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Section: Applications To Representation Theory For O Nmentioning

confidence: 99%

QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

Kribs¹

2003

Proceedings of the Edinburgh Mathematical Society

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“…Previously there were known no such clear-cut invariants that classified wavelet families. But the particular wavelets from [BEJ97] …”

Section: Some Main Results In the Papermentioning

confidence: 99%

A geometric approach to the cascade approximation operator for wavelets

Jørgensen

1999

Integr equ oper theory

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A GEOMETRIC APPROACH TO THE CASCADE APPROXIMATION OPERATOR FOR WAVELETS1 Palle E. T. Jorgensen This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically the cascade algorithm in wavelet theory. Let H be a Hilbert space, and let π be a representation of L ∞ (T) on H. Let R be a positive operator in L ∞ (T) such that R (11) = 1 1, where 1 1 denotes the constant function 1. We study operators M on H (bounded, but noncontractive) such thatwhere the * refers to Hilbert space adjoint. We give a complete orthogonal expansion of H which reduces π such that M acts as a shift on one part, and the residual part is H (∞) = n [M n H], where [M n H] is the closure of the range of M n . The shift part is present, we show, if and only if ker (M * ) = {0}. We apply the operator-theoretic results to the refinement operator (or cascade algorithm) from wavelet theory. Using the representation π, we show that, for this wavelet operator M, the components in the decomposition are unitarily, and canonically, equivalent to spaces L 2 (E n ) ⊂ L 2 (R), where E n ⊂ R, n = 0, 1, 2, . . . , ∞, are measurable subsets which form a tiling of R; i.e., the union is R up to zero measure, and pairwise intersections of different E n 's have measure zero. We prove two results on the convergence of the cascade algorithm, and identify singular vectors for the starting point of the algorithm. Contents:1 In wavelet theory, one is given a subband filter, i.e., a function m 0 on the unit circle, satisfying (i)-(iii) from below and one wants to construct a scaling function ϕ (relative to m 0 ), i.e., a nonzero function ϕ on R satisfying the scaling relation ϕ = Mϕ (relation (1.1) in the paper). Here M = M m 0 is the so-called cascade operator defined bywhere a n are Fourier coefficients of the function m 0 . The scaling function ϕ is important because its shifts generate (under some analytic conditions) the so-called multiresolution subspace V = V (ϕ), which is used to construct the wavelets. There are several ways of constructing a scaling function ϕ. The cascade algorithm is one of the possibilities. In this algorithm one picks some simple function h and considers its iterations M n h. Clearly, if the iterations M n h converge to a nonzero function ϕ, the function ϕ satisfies ϕ = Mϕ, so the algorithm gives us a scaling function. We study the problem of convergence of this algorithm in the setting of an abstract Hilbert space. The cascade operator M has some very special structure. The Ruelle operator R, which appears naturally in this type of problem, gives us a way to describe this structure. Namely, the operator M is what we shall call a sub-isometry; see Definition 6.1. In fact the concept depends on a pair (R, π) where R is a Ruelle operator and π is a representation. It turns out that sub-isometries admit an analogue of the Kolmogorov-Wold decomposition for usual isometries. Using this decomposition, we obtain results about convergence of the cascade algorithm in an abstract ...

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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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Compactly Supported Wavelets and Representations of the Cuntz Relations

Cited by 35 publications

References 25 publications

QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

A geometric approach to the cascade approximation operator for wavelets

Geometric aspects of frame representations of abelian groups

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