Let $\phi$ be a positive definite radial kernel in $\mathbb{R}^d$. I have a point cloud with positions $(\mathbf{x}_i)_{i\in I}$. RBF interpolation of real valued data $f = (f_i)_{i\in I}$ on this cloud would read $\hat{f}(\mathbf{x})=\sum_{i\in I} w_i \phi(\mathbf{x}-\mathbf{x}_i)$, where the RBF weights $g_i$ are solution of the following system : $$ \Phi w = f $$ Where $\Phi_{i,j} = \phi(\mathbf{x}_j-\mathbf{x}_i)$. This matrix is positive definite because $\phi$ itself is, regardless of the positions of the points.
The function I want to interpolate is itself a RBF, but centered at a point $\mathbf{y}$ which is not in the point cloud : $\forall i \in I, \mathbf{y} \neq \mathbf{x}_i$ $$ f_i = \phi(\mathbf{y}-\mathbf{x}_i)$$ My question is the following : I would like to prove that the associated weights are non-negative. (I am not sure it is true but I have not found a counterexemple yet).
Using Farkas lemma, proving this reduces to proving the following : $$\left( \forall j \in I, \sum_{i\in I} \phi(\mathbf{x}_j-\mathbf{x}_i)f_i \ge 0 \right) \quad \Rightarrow \quad \sum_{i\in I} \phi(\mathbf{y}-\mathbf{x}_i)f_i \ge 0$$ Any thoughts ?
I tried it numerically, and the property I wanted to prove is actually wrong. Too bad.