Simultaneous Approximation of a Multivariate Function and its Derivatives by Multilinear Splines
In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We obtain the exact uniform approximation error on classes of functions with moduli of continuity bounded above by certain majorants.
Anderson, R., Babenko, Y., & Leskevych, T. (2014). Simultaneous approximation of a multivariate function and its derivatives by multilinear splines. Journal of Approximation Theory, 183, 82-97.