Geodetic and meteorological 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 Rq, we ...

For K ⊂ C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on C\K are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K) > 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) < 1. Thus only the case when dim(K) = 1 is interesting. So far there is no characterization of γ(K) = 0 in general, but...

The purpose of the present paper is to prove the following result: Let F(P), P= (xi, x2, • ■ • , x„), be a function harmonic for xn >0 and such that for every point Q of a set £ of positive measure on the hyperplane x„ = 0 there exists a region contained in x„>0 limited by a cone with vertex at Q and a hyperplane x„ = const, where the function is bounded. Then except for a set of measure Zero, ...