Date: Dec 1, 2011 7:34 PM
Author: Brad Cooper
Subject: Q: First-Degree Spline Accuracy Theorem - Cheney & Kincaid


In Numerical Mathematics and Computing by Cheney & Kincaid they give the

First-Degree Spline Accuracy Theorem and then say...

if f ' and f '' exist and are continuous, then more can be said, namely

| f(x) - p(x) | <= M1*h/2 (a <= x <= b)

| f(x) - p(x) | <= M2*h^2/8 (a <= x <= b)

I have not been able to derive these last two results. I have tried to use
Taylors Theorem and the Mean Value Theorem, but I am not getting anywhere.

Any help much appreciated.

Here is the First-Degree Spline Accuracy Theorem:

Let p be a first-degree spline having knots a = x0 < x1 < ... < xn = b.
If p interpolates a function f at these knots, then with h = max( x_i -
x_(i-1) )
we have

| f(x) - p(x) | <= w(f;h) (a <= x <= b)

w(f;h) is the modulus of continuity of f