This report continues the study of approximation by bivariate smooth splines on a three-direction mesh. Initiated by de Boor, DeVore and Hollig, box splines have proved useful in determining the approximation order from certain spaces of bivariate splines. By using box splines, de Boor and Hollig gave a sharp upper bound for the approximation order, and Jia got a sharp lower bound for it. But there is still a gap between these two bounds. While determining the exact value of the approximation order is still a formidable problem, Dahmen and Micchelli consider the so-called controlled approximation order from certain spaces of bivariate splines. In their study, Dahmen and Micchelli use a characterization result of Strang and Fix concerning controlled approximation. However, the result of Strang and Fix has been shown to be not true in their original sense. After adjusting the definition of controlled approximation order suitably, in another report, the author obtains the desired characterization property for controlled approximation by box splines. Hereafter he shall refer to controlled approximation in the latter sense. This report determines completely the controlled approximation order from the span of all box splines of any given order and smoothness.
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