On spherical averages of radial basis functions (Q941799)

First of all the author proves that the spherical average operator \(A,\) \[ Af(x)=\int_{S^{d-1}}f(\| x\|\theta)d\mu(\theta),\quad x\in\mathbb R^d, \] (where \(\|\cdot\|\) denotes the Euclidean norm and \(\mu\) is the probability \((d-1)\)-dimensional Lebesgue measure on the unit sphere \((S^{d-1})\) and the Fourier transform operator \(F\) commute when applied to elements of the vector space \(S(\mathbb R^d)\) of infinitely differentiable real-valued functions whose derivative has supra-algebraic decay and to elements of its dual. Then it is proved that the spherical average of radial basis functions \(s(x)=\sum_{k=1}^na_k\phi(x-b_k),\) is compactly supprted if and only if \[ \sum_{k=1}^na_k\| b_k\|^{2l}=0,\quad\mathrm{ for }l=0,1,\ldots,n-1. \] Here \(\phi\) is a polyharmonic spline, i.e. \(\phi\in S(\mathbb R^d)\) and such that its Fourier transform takes the form \(\hat{\phi}(\xi)=C\|\xi\|^{-2m},\xi\in\mathbb R^{d}\backslash\{0\},m\in\mathbb N.\) That result is applied to the special cases of the Euclidian norm \(\phi(x)=\| x\| ,\) for \(x\in\mathbb R^d\) and \(d\) odd, and the thin plate spline \(\phi(x)\| x\|^2\log_e\| x\|,\) for \(x\in\mathbb R^d\) and \(d\) even.