In this paper, the construction of $C^{1}$ cubic quasi-interpolants on a three-direction mesh of $\RR^{2}$ is addressed. The quasi-interpolating splines are defined by directly setting their Bernstein-B\'{e}zier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree. Moreover, additional global properties are required. Finally, we provide some numerical tests confirming the approximation properties.

翻译：本文研究了$\RR^{2}$三维方向网格上$C^{1}$三次拟插值函数的构造问题。通过直接基于点值和梯度值设定每个三角形对应的Bernstein-Bézier系数，所定义的拟插值样条能够复现尽可能高次的多项式。此外，本文还提出了额外全局性质的要求。最后，通过数值实验验证了其逼近特性。