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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 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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See e.g. L. Schwartz, Théorie des Distributions, Example 2.3.2, for a fundamental solution of the $m$-times interated Laplaceoperator in $R^n$, i.e. some S \in D' with $\Delta^m S = \delta$. It is $S(x) = C_{m,n} ||x||^{2m-n}$ if $2m-n$ is odd and $S(x) = C_{m,n} ||x||^{2m-n} \log(||x||)$ in the other case.

Apply this to your specific $\phi$ and check the constants.

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    Do you know the importance of a "fundamental solution"? If $P$ is a differential operator and $S \in D'$ a solution of $PS=\delta$, then $S \ast f$ is a solution of $PS = f$ (if $f$ is reasonable).2011-04-16