Means, Generalized Divided Differences, and Intersections of Osculating Hyperplanes

Divided Differences

Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences. “It belongs to the most beautiful I have been able to do.”[Newton 1676]

Age-related differences in visual scanning at median-divided highway intersections in rural areas.

The objective of this study was to examine age-related differences in visual scanning as drivers performed three separate maneuvers (going straight across, making a left and right turn) at two median-divided highway intersections with different crash frequencies. An on-road study was conducted with 60 drivers in three age groups: younger (18-25), middle-aged (35-55), and older (65-80). The stud...

Interpolating polynomials and divided differences

We shall take it as known that a polynomial of degree n has at most n distinct zeros (a proof is given in Lemma 1 below). Given n+1 distinct real numbers xj and any numbers αj (0 ≤ j ≤ n), there is a unique polynomial p of degree at most n satisfying p(xj) = αj (0 ≤ j ≤ n). The polynomial is unique, since if p1 and p2 were two such polynomials, then p1−p2 would be zero at each xj: since it has ...

Generalized Steffensen Means

Divided differences of implicit functions

Divided differences can be viewed as a discrete analogue of derivatives and are commonly used in approximation theory, see [1] for a survey. Recently, the second author and Lyche established two univariate chain rules for divided differences [2], both of which can be viewed as analogous to Faà di Bruno’s formula for differentiating composite functions [6,7]. One of these formulas was simultaneo...