Discrete Wavelet Frames on the sphere

In this paper we exploit the Continuous Wavelet Transform (CWT) on the sphere introduced in [1, 2] to build the associated Discrete Wavelet Frames. We first explore half-continuous frames, i.e, frames where the position remains a continuous variable, and then move on to a fully discrete theory. This forces us to introduce the notion of controlled frames [5], which reflects the particular nature of the underlying theory, particularly the apparent conflict between dilation and the compacity of the S² manifold. We conclude with some numerical illustrations and future work.