Let us consider some interpolation problems:
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We have some points $n$ points $x_i \in \mathbb R^d$ along with corresponding values $y_i \in \mathbb R$. We'd like to find a function $f: \mathbb R^d \to \mathbb R$ subject to the constraints $f(x_i) = y_i \forall i=1, \ldots, n$.
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For $d=1$ if we additionally require $f$ to minimize the bending energy $E = \int \left(\frac{d^2f}{dx^2} \right) dx$ (the energy of a thin rod bent in a way such that it goes through $(x_i, y_i)$), then it turns out $f$ must be the cubic spline. We can represent cubic splines using the RBF (radial basis function) $\phi(r) = \vert r \vert^3$.
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For $d=2$ we can repeat the same, this time instead of a rod, we use a thin idealized piece of e.g. sheet metal, where the bending energy is $E = \int \left(\frac{\partial^2 f}{\partial x^2}\right)^2 + 2\left(\frac{\partial^2 f}{\partial x\partial y}\right)^2+\left(\frac{\partial^2 f}{\partial y^2}\right)^2 dxdy$. We can then represent the function $f$ minimizing $E$ using thin plate splines which can be represented using the RBF $\phi(x) = |r|^2 \log(|r|)$.
Are there any $RBFs$ known for the solutions of when we generalize this problem to $d=3$ and $d>3$?
I'm not sure but I assume in the general case the energy would be $$E = \int \sum\limits_{i_1 + i_2 + \ldots+ i_n = 2} \binom{2}{i_1,i_2,\ldots,i_n}\left( \frac{\partial^2f}{\prod_j \partial x_{j}^{i_j}} \right)^2 d\pmb x.$$
Best Answer
Indeed: Polyharmonic splines solve this problem, they are defined as
$$\varphi(x) = \begin{cases} r^k & k=1,3,5, \ldots \\ r^k \log r & k=2,4,8,\ldots\end{cases}$$
see https://en.wikipedia.org/wiki/Polyharmonic_spline#Polyharmonic_smoothing_splines