Quasi-interpolation is one method of generating approximations from a space of translates of dilates of a single function psi. This method has been applied widely to approximation by radial basis functions. However, such analysis has most often been performed in the setting of an infinite uniform grid of centers. In this paper we develop general error bounds for approximation by quasi-interpolation on an n-cube. The quasi-interpolant analyzed involves a finite number, growing as h(-n), of translates of dilates of the function psi, and a bounded number of edge functions. The centers of the translates of dilates of psi form a uniformly spaced grid within the cube. These error bounds are then applied to approximation by thin-plate splines on a square. The result is an O(omega(f, [-1, 1]2, h)) error bound for approximation by thin-plate splines supplemented with eight arctan functions.
