Error bounds in divided difference expansions. A probabilistic perspective

We give error bounds for the remainder term, as well as sharp upper and lower bounds for the leading term, in divided difference expansions. A main feature of this paper is the use of a probabilistic approach in the spirit of Ignatov and Kaishev [Z.G. Ignatov, V.K. Kaishev, B-splines and linear combinations of uniform order statistics, Math. Res. Cent. TSR, 2817, Univ. of Wisconsin, Madison, 1985; Z.G. Ignatov, V.K. Kaishev, A probabilistic interpretation of multivariate B-splines and some applications, Serdica 15 (1989) 91–99] and Karlin et al. [S. Karlin, C.A. Micchelli, Y. Rinott, Multivariate splines: A probabilistic perspective, J. Multivariate Anal. 20 (1986) 69–90] based on the representation of divided differences as mathematical expectations involving linear combinations of uniform order statistics.