Multipoint Taylor formulae

The main result of this paper is an abstract version of the Kowalewski - Ciarlet - Wagschal multipoint Taylor formula for representing the pointwise error in multivariate Lagrange interpolation. Several applications of this result are given in the paper. The most important of these is the construction of a multipoint Taylor error formula for a general finite element, together with the corresponding L-p-error bounds. Another application is the construction of a family of error formulae for linear interpolation (indexed by real measures of unit mass) which includes some recently obtained formulae. It is also shown how the problem of constructing an error formula for Lagrange interpolation from a D-invariant space of polynomials with the property that it involves only derivatives which annihilate the interpolating space can be reduced to the problem of finding such a formula for a 'simpler' one-point interpolation map.