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

P. Cerejeiras; M. Ferreira; U. Kähler; F. Sommen

doi:10.3934/cpaa.2007.6.619

Article Contents

2007, Volume 6, Issue 3: 619-641. Doi: 10.3934/cpaa.2007.6.619

This issue Previous Article Spectral Galerkin method for stochastic wave equations driven by space-time white noise Next Article Fast numerical collocation solutions of integral equations

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

1.
Department of Mathematics, University of Aveiro, P-3810-193 Aveiro

Received: February 2006

Revised: September 2006

Published: August 2007

Abstract Related Papers Cited by

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 Möbius 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.

Keywords:

Mathematics Subject Classification: Primary: 42C40, Secondary: 30G35.

Citation:

Article Metrics

HTML views() PDF downloads(87) Cited by(0)

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

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

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

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content