A FAMILY OF IDEALS OF MINIMAL REGULARITY AND THE HILBERT SERIES OF C-R((DELTA)OVER-CAP)

For a simplicial subdivision Delta of a region in R-2, We analyze the dimension of the vector space C-k(r)(Delta) of C-r piecewise polynomia l functions (splines) on Delta of degree at most k. We find an exact s equence which allows us to prove that the dimension series for splines given by Billera and Rose [Discrete Comput. Geom. 6 (1991), 107-128] does indeed agree with the bounds on the dimension of the spline space given by Alfeld and Schumaker [Constr. Approx. 3 (1987), 189-197; Num er. Math. 57 (1990), 651-661]. We give sufficient conditions for the A lfeld-Schumaker bounds to be attained in all degrees, where Delta is a two-dimensional simplicial complex. The conditions are satisfied by t he class of complexes considered by Chui and Wang [J. Math. Anal. Appl . 94 (1983), 197-221], but also by a much broader class of complexes. Furthermore, for conditions which involve only local geometric data, t his result is the strongest possible. (C) 1997 Academic Press.