INTERPOLATION ON SPHERICAL DOMAINS

Let S denote the (to + l)-dimensional unit sphere and let S°° denote the real Hilbert sphere. Given distinct and not pairwise antipodal points ... ,xn on some S, we investigate the possibility of interpolating arbitrary data at the x; by a function of the form χ —• £ " = 1 Cjf(dm(x,xj)), when / is a conditionally positive definite function on S. Here, dm denotes the great-circle distance on S. AMS(MSO) subject classifications: 41A05, 41A58, 65D05 I N T R O D U C T I O N . We denote by S the unit sphere in IR + 1 and by 5 0 0 the unit sphere in any real separable infinite dimensional Hilbert space (this space is denoted by IR). These spaces are metric under the geodesic or great-circle distance dm: dm(x,y) = arccos(x,y)m, x,y € S. Here (·, -)m denotes the usual inner product in IR. Let S denote a subset of some S and let D(S,dm) be the set of all numbers of the form dm(x, y) where χ and y are elements of S. A continuous function / : D(S,dm) —• IR is said to be conditionally positive definite on S if, for any positive integer η and for any set of η points x i , x 2 , . . . ,xn on S, the η χ η matrix A having entries A,j = f(dm(x,,xj)) is almost nonnegative 1 Partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnologico CNPq (Brazil)