University of Georgia

Mathematics Department Colloquium

Prof. Joe Ward

Texas A&M University

November 9, 2000

Representing and analyzing scattered data on spheres

Abstract: Geodetic and meteorlogical data, collected via satellites for example, are genuinely scattered, and not confined to any special set of points. Even so, known quadrature formulas used in numerically computing integrals involving such data have had restrictions either on the sites (points) used or, more significantly, on the number of sites required. Here, for the unit sphere embedded in $R^n$ , we obtain quadrature formulas that are exact for spherical harmonics of a fixed order, have nonnegative weights, and are based on function values at scattered sites. To be exact, these formulas require only a number of sites comparable to the dimension of the space. As a part of the proof, we derive $L_1$-Marcinkiewicz-Zygmund inequalities for such sites.