AbstractLet $\sOm$ be the closure of a bounded open set in $\mathbb{R}^d$, and, for a sufficiently large integer $\kappa$, let $f\in C^\kappa(\sOm)$ be a real-valued ‘bump’ function, i.e. $\supp(f)\subset\textint(\sOm)$. First, for each $h>0$, we construct a surface spline function $\sigma_h$ whose centres are the vertices of the grid $\mathcal{V}_h=\sOm\cap h\zd$, such that $\sigma_h$ approximates $f$ uniformly over $\sOm$ with the maximal asymptotic accuracy rate for $h\rightarrow0$. Second, if $\ell_1,\ell_2,\dots,\ell_n$ are the Lagrange functions for surface spline interpolation on the grid $\mathcal{V}_h$, we prove that $\max_{x\in\sOm}\sum_{j=1}^n\ell_j^2(x)$ is bounded above independently of the mesh-size $h$. An interesting consequence of these two results for the case of interpolation on $\mathcal{V}_h$ to the values of a bump data function $f$ is obtained by means of the Lebesgue inequality.AMS 2000 Mathematics subject classification: Primary 41A05; 41A15; 41A25; 41A63
Range of the gradient of a smooth bump function in finite dimensions
