UNIVERSITY OF WISCONSIN-MADISON CENTER FOR THE MATHEMATICAL SCIENCES A characterization of the approximation order of multivariate spline spaces

We analyze the approximation order associated with a directed set of spaces, {Sh}h>0, each of which spanned by the hZZ-translates of one compactly supported function φh : IR s → C. Under a regularity condition on the sequence {φh}h, we show that the optimal approximation order (in the ∞-norm) is always realized by quasi-interpolants, hence in a linear way. These quasi-interpolants provide the best approximation rates from {Sh}h to an exponential space of good approximation order at the origin. As for the case when each Sh is obtained by scaling S1, under the assumption ∑ α∈ZZ φ1(· − α) 6≡ 0, (∗) 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. AMS (MOS) Subject Classifications: primary 41A15, 41A25, 41A40, 41A63; secondary 65D15.