Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis

Abstract

In this paper we construct a continuous wavelet transform (CWT) on the sphere $S^{n-1}$ based on the conformal group of the sphere, the Lorentz group Spin$(1,n)$. For this purpose, we present a short survey on the existing techniques of continuous wavelet transform and of conformal transformations on the unit sphere. We decompose the conformal group into the maximal compact subgroup of rotations Spin$(n)$ and the set of Mobius transformations of the form $\varphi_a(x) = (x-a)(1+ax)^{-1}$, where $a \in B^n$ and $B^n$ denotes the unit ball in $\mathbb R^n$. Based on a study of the influence of the parameter $a$ arising in the definition of dilations/contractions on the sphere we define a class of local conformal dilation operators and consequently a family of continuous wavelet transforms for the Hilbert space of square integrable functions on the sphere $L_2(S^{n-1})$ and the Hardy space $H^2$. In the end we construct Banach frames for our wavelets and prove Jackson-type theorems for the best $n$-point approximation.