Wavelet subspaces invariant under groups of translation operators (original) (raw)
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Monatshefte für Mathematik, 2010
The wavelet subspaces of the space of square integrable functions on the affine group with respect to the left invariant Haar measure dν are studied using the techniques from [9] with respect to wavelets whose Fourier transforms are related to Laguerre polynomials. The orthogonal projections onto each of these wavelet subspaces are described and explicit forms of reproducing kernels are established. Isomorphisms between wavelet subspaces are given.
On the structure of the space of wavelet transforms
Comptes Rendus Mathematique, 2008
Let G be the "ax + b"-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on L2(R). The complete decomposition of L2(G, dν) onto the space of wavelet transforms W ψ (L2(R)) is obtained after identifying the group G with the upper half-plane Π in C. To cite this article: O. Hutník, C. R. Acad. Sci. Paris, Ser. I 346 (2008).
Wavelet Decompositions of Nonrefinable Shift Invariant Spaces
Applied and Computational Harmonic Analysis, 2002
The motivation for this work is a recently constructed family of generators of shift-invariant spaces with certain optimal approximation properties, but which are not refinable in the classical sense. We try to see whether, once the classical refinability requirement is removed, it is still possible to construct meaningful wavelet decompositions of dilates of the shift invariant space that are well suited for applications. 1 2 1 J J fS −+ − Φ ∈Φ , ( ) 1 2 1 J J fS −+ − Ψ ∈Ψ . Then, 1 J f − Φ plays the role of a low resolution approximation to J f Φ , while 1 J f − Ψ is the difference between the two, the detail. Typically, if J f Φ is a sufficiently smooth function or J is sufficiently large, then 1 JJ ff − ΦΦ ≈ and 1 2 3 2222 JJJJ SSSS −−+−+−+ Φ=Ψ+Ψ+Ψ+ L, (1.4) i.e., any ( ) 2 J J fS − Φ ∈Φ possesses a decomposition 123 JJJJ ffff −−− ΦΨΨΨ =+++ L .
A functional Hilbert space approach to the theory of wavelets
Autom Constr, 2004
We approach the theory of wavelets from the theory of functional Hilbert spaces. Starting with a Hilbert space H, we consider a subset V of H, for which the span is dense in H. We define a function of positive type on the index set I which labels the elements of V. This function of positive type induces uniquely a functional Hilbert space, which is a subspace of C I and there exists a unitary mapping from H onto this functional Hilbert space. Such functional Hilbert spaces, however, are not easily characterized. Next we consider a group G for the index set I and create the set V using a representation R of the group on H. The unitary mapping between H and the functional Hilbert space is easily recognized as the wavelet transform. We do not insist the representation to be irreducible and derive a generalization of the wavelet theorem as formulated by Grossmann, Morlet and Paul. The functional Hilbert space can in general not be identified with a closed subspace of L 2 (G), in contrast to the case of unitary, irreducible and square integrable representations. Secondly, we take for G a semi-direct product of two locally compact groups S T , where S is abelian. In this case we give a more tangible description for the functional Hilbert space, which is easier to grasp. Finally, we provide an example where we take H = L 2 (R 2) and the Euclidean motion group for G. This example is inspired by an application of biomedical imaging, namely orientation bundle theory, which was the motivation for this report.
Wavelets invariant under finite reflection groups
Mathematical Methods in The Applied Sciences, 2009
In this paper we use approximate identities in the Dunkl setting in order to construct spherical Dunkl wavelets, which do not involve the knowledge of the intertwining operator, the Dunkl translation or of the Dunkl transform. The practicality of the proposed approach will be shown with the example of Abel-Poisson wavelets. Copyright © 2009 John Wiley & Sons, Ltd.
An Equivalence Relation on Wavelets in Higher Dimensions
Bulletin of the London Mathematical Society, 2004
We introduce an equivalence relation on the set of single wavelets of L 2 (R n) associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of wavelets and it is shown that each of these classes is non-empty. 2000 Mathematics Subject Classification. 42C40.
Images of the continuous wavelet transform
Contemporary Mathematics, 2014
A wavelet, in the generalized sense, is a vector in the Hilbert space, Hπ, of a unitary representation, π, of a locally compact group, G, with the property that the wavelet transform it defines is an isometry of Hπ into L 2 (G). We study the image of this transform and how that image varies as the wavelet varies. We obtain a version of the Peter-Weyl Theorem for the class of groups for which the regular representation is a direct sum of irreducible representations.
The dilation property for abstract Parseval wavelet systems
In this work we introduce a class of discrete groups called wavelet groups that are generated by a discrete group Γ0 (translations) and a cyclic group Γ1 (dilations), and whose unitary representations naturally give rise to a wide variety of wavelet systems generated by the pseudo-lattice Γ = Γ1Γ0. We prove a condition in order that a Parseval frame wavelet system generated by Γ can be dilated to an orthonormal basis that is also generated by Γ via a super-representation. For a subclass of groups where Γ0 is Heisenberg, we show that this condition always holds, and we cite a number of familiar examples as applications.