Bounds on Projections onto Bivariate Polynomial Spline Spaces with Stable Bases

We derive L 1 bounds for norms of projections onto bivariate polynomial spline spaces on regular triangulations with stable local bases. We then apply this result to derive error bounds for best L 2-and`2-approximation by splines on quasi-uniform triangulations. x1. Introduction Let X L 1 (() be a linear space deened a set with polygonal boundary. Suppose hh; i is a semi-deenite inner-product on X with associated semi-norm k k. We assume that hf; gi = 0, whenever fg = 0 on , (1:1) kfk kgk, whenever jf(x)j jg(x)j for all x 2 : (1:2) Suppose S X is a linear space of polynomial splines (bivariate piecewise poly-nomials) deened on a regular triangulation 4 of (two triangles intersect only at a common vertex or along a common edge). We assume that S is a Hilbert space with respect to hh; i.