Multivariate Hermite interpolation by algebraic polynomials: A survey

This is a survey of that theory of multivariate Lagrange and Hermite interpolation by algebraic polynomials, which has been developed in the past 20 years. Its purpose is not to be encyclopedic, but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples. It takes “classical” Hermite interpolation as a starting point, but then successively broadens the assumptions so that, finally, interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed.