Mean value theorem (divided differences)

In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.[1]

Statement of the theorem

For any n + 1 pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point

 \xi \in (\min\{x_0,\dots,x_n\},\max\{x_0,\dots,x_n\}) \,

where the nth derivative of f equals n ! times the nth divided difference at these points:

 f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}.

For n = 1, that is two function points, one obtains the simple mean value theorem.

Proof

Let P be the Lagrange interpolation polynomial for f at x0, ..., xn. Then it follows from the Newton form of P that the highest term of P is f[x_0,\dots,x_n](x-x_{n-1})\dots(x-x_1)(x-x_0).

Let g be the remainder of the interpolation, defined by g = f - P. Then g has n+1 zeros: x0, ..., xn. By applying Rolle's theorem first to g, then to g', and so on until g^{(n-1)}, we find that g^{(n)} has a zero \xi. This means that

 0 = g^{(n)}(\xi) = f^{(n)}(\xi) - f[x_0,\dots,x_n] n!,
 f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}.

Applications

The theorem can be used to generalise the Stolarsky mean to more than two variables.

References

  1. de Boor, C. (2005). "Divided differences". Surv. Approx. Theory 1: 4669. MR 2221566.
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