On the asymptotics of Fekete-type points for univariate radial basis interpolation

[摘要] Suppose that K subset of R-d is compact and that we are given a function f is an element of C(K) together with distinct points x(i) is an element of K, 1 less than or equal to i less than or equal to 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 - x(j)\) which interpolates f at the given points. Specifically, we took for coefficients c(j) is an element of R such that [GRAPHICS] has the property that F(x(i)) = f (x(i)), 1 less than or equal to i less than or equal to n. The Fekete-type points of this process are those for which the associated interpolation matrix [g(\x(i) - x(j)\)](l less than or equal toi,jless than or equal ton) 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. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-12-01