Radial kernels, polynomial augmentation, and site geometry

Fix an integer $d\ge 1$. An admitted domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is a nonempty bounded Lipschitz domain (its boundary is locally the graph of a Lipschitz function, with $\mathrm{\Omega }$ on one side) with fixed $r_0>0$ and $0<\theta <\pi \mathrm{/}2$ such that for every $x\in \bar{\mathrm{\Omega }}$ there is an orthogonal map $Q_x$ for which $x+Q_x\{z\in {\mathbb{R}}^d:0<\mathrm{\mid }z\mathrm{\mid }<r_0,z_1>\mathrm{\mid }z\mathrm{\mid }\cos \theta \}\subset \mathrm{\Omega }\mathrm{.}$ An orthogonal map is a real linear map with $Q_x^{\mathsf{T}}{Q}_x=I$, equivalently one preserving the Euclidean inner product. The map $Q_x$ may depend on $x$; no simultaneous selection of the maps is part of this definition.

A real radial kernel on ${\mathbb{R}}^d$ has the form $K(x,y)=\varphi (\mathrm{\mid }x-y\mathrm{\mid })$ for a function $\varphi :[0,\mathrm{\infty })\to \mathbb{R}$. For a finite ordered list of distinct sites $X=(x_1,\dots ,x_N)$, its Gram matrix is $K_X=(K(x_i,x_j){)}_{i,j=1}^N$. The kernel is strictly positive definite if $c^{\mathsf{T}}{K}_{X}c>0$ for every such $X$ and every nonzero $c\in {\mathbb{R}}^N$; the nonstrict positive-semidefinite condition replaces $>$ by $\ge$. Only strict positivity is used for the ordinary interpolation solve below.

For an integer $m\ge 1$, let ${\mathrm{\Pi }}_{m-1}({\mathbb{R}}^d)$ be the real polynomials of total degree at most $m-1$, with an ordered basis $p_1,\dots ,p_J$, where $J=\dim {\mathrm{\Pi }}_{m-1}({\mathbb{R}}^d)$. The evaluation matrix on $X$ is $P_X=(p_j(x_i){)}_{i,j}$. The sites are $m$-unisolvent for this polynomial space when $P_X$ has linearly independent columns. A kernel is strictly conditionally positive definite of order $m$ when, for every finite distinct $X$ and this evaluation matrix, $c^{\mathsf{T}}{K}_{X}c>0$ for every nonzero $c\in {\mathbb{R}}^N$ satisfying $P_X^{\mathsf{T}}c=0$. Thus strict conditional positivity is a strict inequality on the moment-null subspace; it is stronger than the corresponding nonstrict inequality.

For data $y\in {\mathbb{R}}^N$, the strictly positive definite interpolation system is $K_{X}a=y$. For an order-$m$ strictly conditionally positive definite kernel and $m$-unisolvent sites, the polynomially augmented interpolation system is $\left[\begin{array}{cc}{K}_X& P_X\\ P_X^{\mathsf{T}}& 0\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}y\\ 0\end{array}\right]\mathrm{.}$ Whenever the relevant system has a unique solution, its kernel-plus-polynomial function $x\mapsto {\sum }_i{a}_{i}K(x,x_i)+{\sum }_j{b}_j{p}_j(x)$ is called the interpolant. Invertibility for the stated strict hypotheses is supplied by Strict positive and conditionally positive augmented interpolation systems are invertible ↗.

For a finite nonempty site set $X\subset \mathrm{\Omega }$, the fill distance is $h_X={\sup }_{x\in \mathrm{\Omega }}{\min }_{x_i\in X}\mathrm{\mid }x-x_i\mathrm{\mid }$. If $N\ge 2$, the separation radius and mesh ratio are $q_X=\frac{1}{2}{\min }_{i\mathrm{\ne }j}\mathrm{\mid }{x}_i-x_j\mathrm{\mid }$ and ${\rho }_X=h_X\mathrm{/}{q}_X$, respectively. These are exact geometric quantities, independent of any floating-point solver.