Continuous Action of Lie Groups on R and Frames

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 L(R) using the action of closed subgroups H ⊂ GL(n,R) such that H has an open orbit O in R under the action (h, ω) 7→ (h) (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 ∈ O then we construct a frame for the space L O (R) of L-functions whose Fourier transform is supported in O. We apply this to the case where H = H and the stabilizer is a symmetric subgroup, a case discussed for the continuous wavelet transform in [8].