On the approximation power of bivariate quadratic C1 splines

In this paper we investigate the approximation power of local bivariate quadratic C1 quasi-interpolating (q-i) spline operators with a four-directional mesh. In particular, we show that they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds both for the errors and for the spline partial derivatives, in the case the spline is more differentiable than the function. Then such general results are applied to prove new properties of two interesting q-i spline operators, proposed and partially studied in Chui and Wang (Sci. Sinica XXVII (1984) 1129–1142).