The Discrete Approximation Problem for a Special Case of Hermite-Type Polynomial Interpolation

Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants. However, this property is not preserved for the multivariate case. In this paper, the authors first generalize the result of P. Gniadek. As an application, the authors consider the discrete approximation problem for a special case when the interpolation condition contains all partial derivatives of order less than n and one nth order differential polynomial. In addition, for the case of n ≥ 3, the authors use the concept of Cartesian tensors to give a sufficient condition to find a sequence of discrete points, such that the Lagrange interpolation problems at these points converge to the given Hermite-type interpolant.