On the dimension of bivariate spline spaces of smoothnessr and degreed=3r+1

Summary We consider the well-known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally. To date, dimension formulae for such spaces have been established only whend3r+2, (except for the special case wherer=1 andd=4). In this paper we establish dimension formulae for allr1 andd=3r+1 for almost all triangulations.Dedicated to R. S. Varga on the occasion of his sixtieth birthdaySupported in part by National Science Foundation Grant DMS-8701121Supported in part by National Science Foundation Grant DMS-8602337