Continuous wavelet transform on the hyperboloid

In this paper we build a continuous wavelet transform (CWT) on the upper sheet of the 2-hyperboloid . First, we define a class of suitable dilations on the hyperboloid through conic projection. Then, incorporating hyperbolic motions belonging to SO0(1,2), we define a family of axisymmetric hyperbolic wavelets. The continuous wavelet transform Wf(a,x) is obtained by convolution of the scaled axisymmetric wavelets with the signal. The wavelet transform is proved to be invertible whenever wavelets satisfy a particular admissibility condition, which turns out to be a zero-mean condition. We then provide some basic examples and discuss the limit at null curvature.