On smoothing conditions of multivariate splines

Let $T$ be a $k$-simplex in $\mathbb{R}^s$, where $0\leq k < n$, and let $S_a$ and $S_b$ be two adjacent $s$-simplices with $T = S_a\cap S_b$. Suppose that $F(x)\in C(S_a\cup S_b)$ with $$F(x)\vert S_a = P_n(x),$$ $$F(x)\vert S_b = Q_n(x),$$ where $P_n$ i $Q_n$ are Bezier polynomials in $\mathbb{R}^s$ with total degree n. The conditions, which must be required to function $F$ be in class $C^r$ across $T$, are introduced by C.K. Chui and M. Lai ([3], [4]). In the present note the improvement of those conditions is obtained. As an application, algorithm for computation of polynomial coefficients is shown.