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According to equation (2.14) of the paper "The Uniform Convergence of Thin Plate Spline Interpolation in Two Dimensions" a radial basis function $\phi(\parallel x \parallel)$ has the property

$$ \nabla^4 \phi( \parallel x \parallel ) = 8 \pi \delta(x) $$

I'd like help proving this statement.

PS: I'm sorry but I'm not sure what are the appropriate tags for this question.

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    You should probably include the definition of the radial basis function in the statement of your question.2011-04-16
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    Definition taken from the Wikipedia entry: a real-valued function whose value depends only on the distance from the origin. (http://en.wikipedia.org/wiki/Radial_basis_function )2011-04-16
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    As stated, this is wrong as any constant function is radial and its derivate is not delta2011-04-16
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    You should edit your question: $\phi$ is a very specific function in the mentioned article, basicly the fundamental solution $\phi(r) = r \log r$ of the bi-Laplace operator in two dimensions.2011-04-16

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