On structure of bivariate spline space of lower degree over arbitrary triangulation (Q939505)

Let \(\Delta \) be a triangulation of a planar polygon region. Let us denote by \(S_{k}^{\mu} (\Delta )\) the space of multivariate splines of degree \(k\) with smoothness order \(\mu \) over \(\Delta \). \textit{L. L. Schumaker} [Rocky Mt. J. Math. 14, 251--264 (1984; Zbl 0601.41034)] proposed a lower bound and an upper bound for the dimension of the space \(S_{k}^{\mu} (\Delta )\). In this paper the upper bound for \(\dim S_{k}^{\mu} (\Delta )\) given by Schumaker is slightly improved via the new vertex coding method. The authors also study the structure of the space \(S_{\mu}^{k}(\Delta) \) over arbitrary triangulation in the cases \(\mu = 1,\, k=2\) and \(3\).