Wavelet Transforms for Semidirect Product Groups with Not Necessarily Commutative Normal Subgroups (original) (raw)
Related papers
Journal of Fourier Analysis and Applications, 2002
Continuous wavelet transforms arising from the quasiregular representation of a semidirect product group G = R k ⋊ H have been studied by various authors. Recently the attention has shifted from the irreducible case to include more general dilation groups H, for instance cyclic (more generally: discrete) or one-parameter groups. These groups do not give rise to irreducible square-integrable representations, yet it is possible (and quite simple) to give admissibility conditions for a large class of them. We put these results in a theoretical context by establishing a connection to the Plancherel theory of the semidirect products, and show how the admissibility conditions relate to abstract admissibility conditions which use Plancherel theory. *
Continuous wavelet transforms and non-commutative Fourier analysis
2010
We discuss continuous wavelet transforms for the semidirect product group of a unimodular (not necessarily commutative) normal subgroup N with a closed subgroup H of Aut(N ), which is a generalization of the wavelet theory for an affine transformation group on a vector space. The operator-valued Fourier transform for N plays a substantial role in the arguments.
Orthogonal wavelets on locally compact Abelian groups
Functional Analysis and Its Applications, 1997
We extend and improve the results of W. Lang (1998) on the wavelet analysis on the Cantor dyadic group C. Our construction is realized on a locally compact abelian group G which is defined for an integer p 2 and coincides with C when p = 2. For any integers p, n 2 we determine a function ϕ in L 2 (G) which 1) is the sum of a lacunary series by generalized Walsh functions, 2) has orthonormal "integer" shifts in L 2 (G), 3) satisfies "the scaling equation" with p n numerical coefficients, 4) has compact support whose Haar measure is proportional to p n , 5) generates a multiresolution analysis in L 2 (G). Orthogonal wavelets ψ with compact supports on G are defined by such functions ϕ. The family of these functions ϕ is in many respects analogous to the well-known family of Daubechies' scaling functions. We give a method for estimating the moduli of continuity of the functions ϕ, which leads to sharp estimates for small p and n. We also show that the notion of adapted multiresolution analysis recently suggested by Sendov is applicable in this situation.
Wavelet Packets on Locally Compact Abelian Groups
2010
The objective of this paper is to construct wavelet packets associated with multiresolution analysis on locally compact Abelian groups. Moreover, from the collection of dilations and translations of the wavelet packets, we characterize the subcollections which form an orthonormal basis for L 2 (G).
Wavelet transforms via generalized quasi-regular representations
Applied and Computational Harmonic Analysis, 2009
The construction of the well-known continuous wavelet transform has been extended before to higher dimensions. Then it was generalized to a group which is topologically isomorphic to a homogeneous space of the semidirect product of an abelian locally compact group and a locally compact group. In this paper, we consider a more general case. We introduce a class of continuous wavelet transforms obtained from the generalized quasi-regular representations. To define such a representation of a group G, we need a homogeneous space with a relatively invariant Radon measure and a character of G.
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.
Multiresolution analysis and Harr-like wavelet bases on locally compact groups
2012
The multiresolution analysis (MRA) on certain non-abelian locally compact groups G is considered. Characterizations for a refinable function to generate an MRA in L2(G) are given. Here, no regularity properties or decay conditions are placed on the scaling functions. MRAs for L2(G) generated by a self-similar tile as a scaling function are shown and Haar-like wavelet bases are constructed. Concrete examples related to Heisenberg group are provided to illustrate the theorems.
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.