On the Asymptotics of Fekete-Type Points for Univariate Radial Basis Interpolation |
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Authors: | L P Bos U Maier |
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Institution: | a Department of Mathematics and Statistics, University of Calgary, Calgary, T2N 1N4, Alberta, Canadaf1;b Mathematisches Institut, Universität Gießen, Arndtstr. 2, 35392, Gießen, Germany |
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Abstract: | Suppose that K
d is compact and that we are given a function fC(K) together with distinct points xiK, 1in. Radial basis interpolation consists of choosing a fixed (basis) function g :
+→
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
such that
has the property that F(xi)=f(xi), 1in. The Fekete-type points of this process are those for which the associated interpolation matrix g(|xi−xj|)]1i,jn 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. |
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Keywords: | |
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