On the approximation power of bivariate splines

We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces S r d (4) with d 3r+2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the L p norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the the smallest angle in the underlying triangulation and the nature of the boundary of the domain. Let be a bounded polygonal domain in IR 2. Given a nite triangulation 4 of , we are interested in spaces of splines of smoothness r and degree d of the form S r d (4) := fs 2 C r (() : sj T 2 P d ; for all T 2 4g; where P d denotes the space of polynomials of total degree at most d. The main result of this paper is the following theorem which states the existence of a quasi-interpolation operator Q m which maps L 1 (() into the spline space S r d (4) in such a way that if f lies in a Sobolev space W m+1 p (() with 0 m d, then Q m f approximates f and its derivatives to optimal order. Theorem 1.1. Fix d 3r + 2 and 0 m d. Then there exists a linear quasi-interpolation operator Q m mapping L 1 (() into S r d (4) and a constant C such that if f is in the Sobolev space W m+1 p (() with 1 p 1, kD x D