General Existence Theorems for Orthonormal Wavelets, an Abstract Approach

Baggett, Larry; Carey, Alan L.; Moran, William; Ohring, Peter

doi:10.2977/prims/1195164793

Cited by 46 publications

(58 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…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%

Generalized Multiresolution Analyses with Given Multiplicity Functions

Baggett

Larsen

Merrill

et al. 2008

J Fourier Anal Appl

View full text Add to dashboard Cite

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.

show abstract

Section: Multiplicity Functions and The Main Theoremmentioning

confidence: 99%

Generalized Multiresolution Analyses with Given Multiplicity Functions

Baggett

Larsen

Merrill

et al. 2008

J Fourier Anal Appl

View full text Add to dashboard Cite

show abstract

“…It does not address the problem of constructing smooth refinable distributions, nor does it address the problem of constructing wavelet bases. However, we note that Bagget, Carey, Moran, and Ohring showed [1], using properties of von Neumann algebras [19], [9], that the existence of stable refinable functions in L 2 (G) implies the existence of orthonormal wavelet bases for L 2 (G). .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Infinite convolution products and refinable distributions on Lie groups

Lawton

2000

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Sufficient conditions for the convergence in distribution of an infinite convolution product µ 1 * µ 2 * . . . of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy T φ = φ where T is a refinement operator constructed from a measure µ and a dilation automorphism A. The existence of A implies G is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset K ⊂ G such that for any open set U containing K, and for any distribution f on G with compact support, there exists an integer n(U , f) such that n ≥ n(U , f) implies supp(T n f ) ⊂ U. If µ is supported on an A-invariant uniform subgroup Γ, then T is related, by an intertwining operator, to a transition operator W on C(Γ). Necessary and sufficient conditions for T n f to converge to φ ∈ L 2 , and for the Γ-translates of φ to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of W to functions supported on Ω := KK −1 ∩ Γ.

show abstract

“…, ψ L ; and one can consider an orthonormal basis for a closed subspace of L 2 (R). There have also been several publications of wavelets in higher dimensions, cf [1,2,3,5,10,12,13,14,21,22,23] to name few. One of the differences in higher dimensions is that we now have many more choices in the sets of dilations and translations.…”

Section: Introductionmentioning

confidence: 99%

Wavelet sets without groups

Dobrescu¹,

Ólafsson²

2006

Integral Geometry and Tomography

View full text Add to dashboard Cite

Abstract. Most of the examples of wavelet sets are for dilation sets which are groups. We find a necessary and sufficient condition under which subspace wavelet sets exist for dilation sets of the form AB, which are not necessarily groups. We explain the construction by a few examples. IntroductionWavelets and frames have become a widely studied tool in mathematics and applied science. One of the obvious questions is the construction of wavelets with given properties. The classical wavelet system on the line is given by a function ψ ∈ L 2 (R), such that the dyadic dilates and translates of ψ form an orthonormal basis for L 2 (R). Thus, {ψ j,n } j,n∈Z with ψ j,n (t) = 2 j/2 ψ(2 j t + n), is an orthonormal basis for L 2 (R). There are several obvious generalizations: One can replace 2 by any integer N ; one can allow several wavelet functions ψ 1 , . . . , ψ L ; and one can consider an orthonormal basis for a closed subspace of L 2 (R). There have also been several publications of wavelets in higher dimensions, cf [1,2,3,5,10,12,13,14,21,22,23] to name few. One of the differences in higher dimensions is that we now have many more choices in the sets of dilations and translations. So, to fix the notation, let D ⊆ GL(n, R) and T ⊆ R n be countable sets. A (D, T )-wavelet is a square integrable function ψ with the property that the set of functionsforms an orthonormal basis for L 2 (R n ). The set D is called the dilation set and the set T is called the translation set. If we replace L 2 (R n ) in the above definition bywe get a (D, T )-subspace wavelet. Here F stands for the Fourier transformWe will often writef for the Fourier transform of f . The most natural starting point is to consider groups of dilations and full rank lattices as translation sets. The simplest examples would then be groups generated by one element D = {a k | k ∈ Z}, see [24] and the reference therein. In [22,23] 1991 Mathematics Subject Classification. 42C40,43A85.

show abstract

General Existence Theorems for Orthonormal Wavelets, an Abstract Approach

Cited by 46 publications

References 5 publications

Generalized Multiresolution Analyses with Given Multiplicity Functions

Generalized Multiresolution Analyses with Given Multiplicity Functions

Infinite convolution products and refinable distributions on Lie groups

Wavelet sets without groups

Contact Info

Product

Resources

About