Radial multiresolution in dimension three

Abstract

We present a construction of a wavelet-type orthonormal basis for the space of radial $L^2$-functions in ${\mathbb{R}}^3$ via the concept of a radial multiresolution analysis. The elements of the basis are obtained from a single radial wavelet by usual dilations and generalized translations. Hereby the generalized translation reveals the group convolution of radial functions in ${\mathbb{R}}^3$. We provide a simple way to construct a radial scaling function and a radial wavelet from an even classical scaling function on $\mathbb{R}$. Furthermore, decomposition and reconstruction algorithms are formulated.

23 pages