Optimal recovery of twice differentiable functions based on symmetric splines

Given values and gradients of a function at a finite set of nodes in Rd, we introduce symmetric spline recovery methods based on local information. We explicitly construct bivariate symmetric interpolating continuous splines on regular triangulations that solve the optimal global recovery problem studied in Babenko et al. (2010) [2] for the class of functions whose second derivatives in any direction are uniformly bounded. We further prove that in contrast to the univariate case, there are no smooth symmetric splines that solve this recovery problem for d>1d>1.