Approximation of full Lagrange functions by local Lagrange functions

Let $G=(\mathcal{V},E)$ be the graph underlying the neighborhood $B(v_0,K)$, let $\widetilde{\mathcal{V}}\subset\mathcal{V}$ be the interpolation set, and let $\chi(\cdot,v_0)$ and $\bar{\chi}(\cdot,v_0)$ denote respectively the full and local Lagrange functions centered at $v_0\in\widetilde{\mathcal{V}}$, with $K>0$. Approximation conjecture. The local Lagrange functions can be made arbitrarily close to the full Lagrange functions with a relatively small value of $K$. This is intended to justify replacing the computationally expensive full Lagrange functions by local ones, but the statement does not specify a quantitative error bound or a precise meaning of “relatively small”; its resolution is therefore unclear.