Continuous action of Lie groups on Rⁿ and Frames

Original · EN

Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this article we discuss the construction of frames for L²(ⁿ) using the action of closed subgroups H⊂ GL(n,R) such that H has an open orbit in ⁿ under the action (h,ω) (h⁻¹)ᵗ(ω). If H has the form ANR, where A is simply connected and abelian, N contains a co-compact discrete subgroup and R is compact containing the stabilizer group of ω∈ then we construct a frame for the space L²(ⁿ) of L²-functions whose Fourier transform is supported in. We apply this to the case where Hᵗ=H and the stabilizer is a symmetric subgroup, a case discussed for the continuous wavelet transform in a paper by Fabec and Olafsson.