On the Unitary Systems Affiliated with Orthonormal Wavelet Theory inn-Dimensions

Ionascu, Eugen J.; Larson, David R.; Pearcy, Carl

doi:10.1006/jfan.1998.3277

Cited by 8 publications

(11 citation statements)

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“…The goal of the present paper is to investigate the existence and uniqueness of symmetric approximations of frames (respectively, symmetric orthogonalizations of bases of Hilbert spaces) {f i } i∈N of Hilbert subspaces K ⊆ H. That means, we look for the existence and uniqueness of normalized tight frames (resp., orthonormal bases) {ν i } i∈N of Hilbert subspaces L ⊆ H such that the sum j∈N ν j − f j 2 is finite and admits the minimum of all finite sums j∈N µ j − f j 2 that might appear for any other normalized tight frame (resp., orthonormal basis) {µ i } i∈N of any other Hilbert subspace of H. We apply the approach of the third author to symmetric approximations of frames in Hilbert spaces by normalized tight frames. We can rely on fundamental work done by David R. Larson and his collaborators Xingde Dai, Deguang Han, E. J. Ionascu and C. M. Pearcy ( [10,13,16]), by A. Aldroubi [3], P. G. Casazza [4], O. Christensen [7,8,9] and J. R. Holub [14,15]. As a result we obtain that a symmetric approximation exists and is always unique if and only if the operator (P − |F |) is Hilbert-Schmidt, where (I − P ) denotes the projection to the kernel of |F |.…”

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confidence: 99%

Symmetric approximation of frames and bases in Hilbert spaces

Frank

Paulsen

Tiballi

2001

Trans. Amer. Math. Soc.

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Abstract. We introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. We prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames. Even in the case of the symmetric orthogonalization of bases, our techniques and results are new. A crucial role is played by whether or not a certain operator related to the initial frame or basis is Hilbert-Schmidt.

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confidence: 99%

Symmetric approximation of frames and bases in Hilbert spaces

Frank

Paulsen

Tiballi

2001

Trans. Amer. Math. Soc.

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“…This is easily worked out, and was shown in detail in [39] in the context of working out a complete theory of unitary equivalence of wavelet systems. Hence the wavelet theories are equivalent.…”

Section: The Expansive-dilation Casementioning

confidence: 99%

Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames

Larson¹

2007

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a collection, or system, of unitary operators. We will describe the operator-interpolation approach to wavelet theory using the local commutant of a system. This is really an abstract application of the theory of operator algebras to wavelet theory. The concrete applications of this method include results obtained using specially constructed families of wavelet sets. A frame is a sequence of vectors in a Hilbert space which is a compression of a basis for a larger space. This is not the usual definition in the frame literature, but it is easily equivalent to the usual definition. Because of this compression relationship between frames and bases, the unitary system approach to wavelets (and more generally: wandering vectors) is perfectly adaptable to frame theory. The use of the local commutant is along the same lines as in the wavelet theory. Finally, we discuss constructions of frames with special properties using targeted decompositions of positive operators, and related problems.

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“…This is easily worked out, and was shown in detail in [18] in the context of working out a complete theory of unitary equivalence of wavelet systems. Hence the wavelet theories are equivalent.…”

Section: One Dimensionmentioning

confidence: 99%

“…Some of the original work was accomplished in the Ph.D. theses of Q. Gu and D. Speegle, when they were together finishing up at Texas A&M. Some significant additional work was accomplished by Speegle and also by others. In [18], with Ionascu and Pearcy we proved that if an nxn real invertible matrix A is not similar (in the nxn complex matrices) to a unitary matrix, then the corresponding dilation operator D A is in fact a bilateral shift of infinite multiplicity. If a dilation matrix were to admit any type of wavelet (or frame-wavelet) theory, then it is well-known that a necessary condition would be that the corresponding dilation operator would have to be a bilateral shift of infinite multiplicity.…”

Section: Non-expansive Dilationsmentioning

confidence: 99%

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Unitary Systems and Wavelet Sets

Larson

Wavelet Analysis and Applications

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Abstract. A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. We will describe an operator-interpolation approach to wavelet theory using the local commutant of a unitary system. This is an application of the theory of operator algebras to wavelet theory. The concrete applications to wavelet theory include results obtained using specially constructed families of wavelet sets. The main section of this paper is section 5, in which we introduce the interpolation map σ induced by a pair of wavelet sets, and give an exposition of its properties and its utility in constructing new wavelets from old. The earlier sections build up to this, establishing terminology and giving examples. The main theoretical result is the Coefficient Criterion, which is described in Section 5.2.2, and which gives a matrix valued function criterion specificing precisely when a function with frequency support contained in the union of an interpolation family of wavelet sets is in fact a wavelet. This can be used to derive Meyer's famous class of wavelets using an interpolation pair of Shannon-type wavelet sets as a starting point. Section 5.3 contains a new result on interpolation pairs of wavelet sets: a proof that every pair of sets in the generalized Journe family of wavelet sets is an interpolation pair. We will discuss some results that are due to this speaker and his former and current students. And we finish in section 6 with a discussion of some open problems on wavelets and frame-wavelets. Mathematics Subject Classification (2000). Primary 46L99; Secondary 42C15, 46H25.

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On the Unitary Systems Affiliated with Orthonormal Wavelet Theory inn-Dimensions

Cited by 8 publications

References 8 publications

Symmetric approximation of frames and bases in Hilbert spaces

Symmetric approximation of frames and bases in Hilbert spaces

Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames

Unitary Systems and Wavelet Sets

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