Treffer: One Estimate for Divided Differences and its Applications: One estimate for divided differences and its applications

For polynomial and piecewise polynomial (spline) interpolation the forming of divided differences of the approximand's values are important both for the error estimates (usually pointwise or uniform error estimates) and for expressing the approximant itself. In this paper, interesting new estimates for the mentioned divided difference are found which are therefore highly relevant for the approximation error inequalities. The authors not only derive the estimate but show the various applications such as Hermite interpolation, where in this case it is relevant to be able to mimick the ocurrence of the derivatives in the data by divided differences with multiple knots. The estimates are valid for sufficiently smooth approximands and are expressed by certain integrals over moduli of smoothness.