Proving that the bi-laplacian of a radial basis function is the dirac delta

Question

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.

Answer 1

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.