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Knot removal with radial function interpolation

by M. Buhmann and A. Le Méhauté

(Report number 1994-09)

Abstract
In this note we study interpolants to $n$-variate, real valued functions from radial function spaces, \ie spaces that are spanned by radially symmetric functions $\varphi(\|\cdot - x_{j} \|_2)$ defined on $\RR^n$. Here $\| \cdot \|_2$ denotes the Euclidean norm, $\varphi : \RR_+ \to \RR$ is a given "radial (basis) function" which we take here to be $\varphi (r) = ( r^2 + c^2)^{\beta /2}$, $-n \leq \beta < 0$, and the $\{x_j \} \subset \RR^n$ are prescribed "centres", or knots. We analyse the effect of removing a knot from a given interpolant, in order that in applications one can see how many knots can be eliminated from an interpolant so that the interpolant remains within a given tolerance from the original one.

Keywords:

@Techreport{BL94_164,
  author = {M. Buhmann and A. Le Méhauté},
  title = {Knot removal with radial function interpolation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1994-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1994/1994-09.pdf },
  year = {1994}
}

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