Multivariate approximation by a combination of modified Taylor polynomials

We study an approximation of a multivariate function f by an operator of the form ∑ i = 1 N T ˜ r [ f , x i ] ( x ) φ i ( x ) , where φ 1 , … , φ N are certain basis functions and T ˜ r [ f , x i ] ( x ) are modified Taylor polynomials of degree r expanded at x i . The modification is such that the operator has highest degree of algebraic precision. In the univariate case, this operator was investigated by Xuli [Multi-node higher order expansions of a function, J. Approx. Theory 124 (2003) 242–253]. Special attention is given to the case where the basis functions are a partition of unity of linear precision. For this setting, we establish two types of sharp error estimates. In the two-dimensional case, we show that this operator gives access to certain classical interpolation operators of the finite element method. In the case where φ 1 , … , φ N are multivariate Bernstein polynomials, we establish an asymptotic representation for the error as N → ∞ .