On the Continuous Wavelet Transform on Homogeneous Spaces

Abstract

Let G be a locally compact group with a compact subgroup H. We define a square integrable representation of a homogeneous space G/H on a Hilbert space [Formula: see text]. The reconstruction formula for G/H is established and as a result it is concluded that the set of admissible vectors is path connected. The continuous wavelet transform on G/H is defined and it is shown that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we obtain a necessary and sufficient condition for the continuous wavelet transform to be onto.

Technical notice

“…In [5], we have studied the squre integrable representations of homogeneous spaces and admissible wavelets for these representations. Here we define representations of homogeneous spaces which will be needed in the difinition of square integrable representations of homogeneous spaces.…”

“…Consider H = {(0, 0, p 2 ) ∈ G}. Thus the representation of G/H is square integrable [5]. By theorem 2.4 the representation of homogeneous space G/H is square integrable if and only if the representationπ of qoutient group G/N is square integrable, where N = kerπ = {(0, −p 2 cotgγ, p 2 ), p 2 ∈ R}.…”

“…The square integrable representations on homogeneous spaces that admit G-invariant measure and relatively invariant measure have been studied in [1,5]. In this manuscript we investigate the relation between square integrable representations of locally compact group G and its homogeneous space G/H, in which H is compact subgroup of G.…”

Homogeneous spaces and square-integrable representations

“…( For more details about admissible wavelets on locally compact groups the reader can be consult with [15,10,12]). The square integrable representations of homogeneous spaces that admit G-invariant measure and relatively invariant measure have been studied in [2,6]. In this manuscript we investigate the relation between square integrable representations of locally compact group G and its homogeneous space G/H, in which H is a compact subgroup of G. To be more precise we need to fix some notations and review some basic concepts ( see also [9,7,14,4]).…”

Admissible wavelets on groups and their homogeneous spaces

“…Such a unital vector ζ is called an admissible vector. The square integrable representations on homogeneous spaces that admit G-invariant measure and relatively invariant measure have been studied in [1,5]. In this manuscript we investigate the relation between square integrable representations of locally compact group G and its homogeneous space G/H, in which H is compact subgroup of G. To be more precise we need to fix some notations and review some basic concepts.…”