Approximation of the derivatives of a function in Lagrange interpolation on low-dimensional simplices (Q828606)

When functions in one or more dimensions are approximated, there are a number of aspects to be considered: (i) from which space we approximate (in particular, when the dimension is more than one, we may wish to choose splines, radial basis functions, etc), (ii) how we approximate, especially interpolation or the very attractive so-called quasi-interpolation, (iii) whether function values of the approximant only are approximated or -- in Hermite-style -- derivatives as well. In all three choices, the resulting convergence orders are of particular interest. In this paper, new results for three and four dimensions are offered, especially when Lagrange interpolation is carried out on simplices. With respect to our question (ii), interpolations, pointwise, are taken, and in (iii) derivatives are approximated. Convergence orders are established, too. As an application, finite element methods are mentioned as well.