A CHARACTERIZATION OF THE APPROXIMATION ORDER OF MULTIVARIATE SPLINE SPACES

We analyze the approximation order associated with a directed set of spaces, {S(h)}h > 0, each of which is spanned by the hZ(s)-translates of one compactly supported function phi-h: R(s) --> C. Under a regularity condition on the sequence {phi-h}h, we show that the optimal approximation order (in the infinity-norm) is always realized by quasi-interpolants, hence in a linear way. These quasi-interpolants provide the best approximation rates from {S(h)}h to an exponential space of good approximation order at the origin. As for the case when each S(h) is obtained by scaling S1, under the assumption [GRAPHICS] the results here provide an unconditional characterization of the best approximation order in terms of the polynomials in S1. The necessity of (*) in this characterization is demonstrated by a counterexample.