Multiresolution analysis by spherical up functions (Q853535)

Let \(\Omega\) be the unit sphere in \({\mathbb R}^3\). In this paper, the idea of the up function is transformed to the spherical context. Originally, up functions were introduced by \textit{V. A. Rvachev } [Russ. Math. Surv. 45, 87--120 (1990); translation from Usp. Mat. Nauk 45, No. 1(271), 77--103 (1990; Zbl 0704.34090)]. Here the authors introduce generalizations of known locally supported kernels and build an infinite convolution with them. This procedure results in a new class of radial basis functions that are \(C^{\infty}\) and (when the support of the building functions is chosen properly) also have local support. By this procedure the authors are able for the first time to formulate a multiresolution analysis of \(L^2(\Omega)\) by the use of locally supported zonal functions. This paper closes with decomposition and reconstruction schemes.