Multivariate approximation by a combination of modified Taylor polynomials

We study an approximation of a multivariate function f by an operator of the form Sigma(N)(i=1) (T$) over tilde (r)[f, x(i)](x)phi(i)(x), where phi(1),..., phi(N) are certain basis functions and (T$) over tilde (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 phi(1),..., phi(N) are multivariate Bernstein polynomials, we establish an asymptotic representation for the error as N -> infinity. (c) 2005 Elsevier B.V. All rights reserved.