Second Differences and Higher Order Smoothness

In Section 1 we derived Leibniz' second derivative formula for the radius of curvature of a curve.  We actually used infinitesimal second differences, rather than second derivatives and a complete justification requires some more work.  We conclude this Section with a result connecting higher order infinitesimal differences and interated derivatives.

One way to re-state the Uniform First Derivative Theorem above is: The curve y = f[x] is smooth if and only if the line through any two pairs of infinitely close points on the curve is near the same real line,

x_1≈x_2 ⇒ (f[x_1] - f[x_2])/(x_1 - x_2) ≈m

A natural way to extend this is to ask: What is the parabola through three infinitely close points?  Is the (standard part) of it independent of the choice of the triple?  In A Discrete Condition for Higher-Order Smoothness, Boletim da Sociedade Portugesa de Matematica, n.35, Outtubro de 1996, p. 81-94, Vitor Neves and I show:

Theorem: Theorem on Higher Order Smoothness

Let f [x] be a real function defined on a real open interval (α, ω). Then f[x] is n-times continuously differentiable on (α, ω) if and only if the n^th-order differences δ^nf are S-continuous on (α, ω).  In this case, the coefficients of the interpolating polynomial are near the coefficients of the Taylor polynomial,[x] on the interval (a, b).”

δ^nf[x _0, ..., x_n] ≈1/n ! f ^(n)[b]

whenever the interpolating points satisfy x_1≈⋯≈x_n ≈b.

For more details see p.108.

Created by Mathematica  (September 22, 2004)