Level sets on spheres

Fundamental sets of functions on spheres

Let S be the unit sphere in R and let 〈 · , · 〉 denote the usual inner product in R. We characterize those functions K in L([−1, 1]), p ≥ 1, for which the associated set of zonal functions {x ∈ S → K(〈x, y〉) : y ∈ S } is fundamental in L(S). We then study fundamentality of sets generated by either spherical convolution or spherical shifting, thus providing methods of construction of fundamental...

Fixed Point Sets of Involutions on Spheres

Norming Sets and Scattered Data Approximation on Spheres

This short note deals with approximation order of spaces spanned by (x;), x 2 X, with a positive deenite kernel on a sphere and X a given set of nodes. We estimate both the L 2-and the L 1-error, if the function to be approximated is assumed to be rather smooth, thus deviating from the usual assumption that the function stems from thèna-tive space' of the kernel. It is of interest that, based o...

Characteristic Fixed-Point Sets of Semifree Actions on Spheres

A group action is semifree if it is free away from its fixed-point set. P. A. Smith showed that when a finite group of order q acts semifreely on a sphere, the fixed set is a mod q homology sphere. Conversely, given a mod q homology sphere as a subset of a sphere, one may try to construct a group action on the sphere fixing the subset. The converse question was first systematically studied by J...

MEAN VALUE INTERPOLATION ON SPHERES

In this paper we consider   multivariate Lagrange mean-value interpolation problem, where interpolation parameters are integrals over spheres. We have   concentric spheres. Indeed, we consider the problem in three variables when it is not correct.