## Continuous Wavelet Transform on the Hyperboloid

In this paper we build a Continuous Wavelet Transform (CWT) on the upper sheet of the 2-hyperboloid $H_+^2$. First, we define a class of suitable dilations on the hyperboloid through conic projection. Then, incorporating hyperbolic motions belonging to $SO_0(1,2)$, we define a family of hyperbolic wavelets. The continuous wavelet transform $W_f(a,x)$ is obtained by convolution of the scaled 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.

Published in:
Applied and Computational Harmonic Analysis, 23, 3, 286-306
Year:
2007
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