On the Asymptotics of Fekete-Type Points for Univariate Radial Basis Interpolation

AbstractSuppose that K⊂Rd is compact and that we are given a function f∈C(K) together with distinct points xi∈K, 1⩽i⩽n. Radial basis interpolation consists of choosing a fixed (basis) function g : R+→R and looking for a linear combination of the translates g(|x−xj|) which interpolates f at the given points. Specifically, we look for coefficients cj∈R such that F(x)=∑j=1ncjg(|x−xj|) has the property that F(xi)=f(xi), 1⩽i⩽n. The Fekete-type points of this process are those for which the associated interpolation matrix [g(|xi−xj|)]1⩽i,j⩽n has determinant as large as possible (in absolute value). In this work, we show that, in the univariate case, for a broad class of functions g, among all point sequences which are (strongly) asymptotically distributed according to a weight function, the equally spaced points give the asymptotically largest determinant. This gives strong evidence that the Fekete points themselves are indeed asymptotically equally spaced