PROOFS OF SELECTED THEOREMS, AND ADDITIONAL MATERIAL
The interpolation error theorems all depend on applying a generalization of Rolle’s Theorem to a very carefully constructed function. First, we will state and prove the general version of Rolle’s Theorem.
Theorem A.1 (Generalized Rolle’s Theorem) Let f Cn([a, b]) be given, and assume that there are n points, zk, 1 ≤ k ≤ n in [a, b] such that f(zk) = 0. Then there exists at least one point ξ [a, b] such that f(n-1)(ξ) = 0.
Proof: By Rolle’s Theorem, there exists at least one point ηk between each zk and zk+1 such that f′(ηk) = 0. Thus there are n − 1 points where g1(x) = f′(x) is zero. By the same argument, then, there are n − 2 points where g2(x) = f″(x) is zero. Continuing onward, then, we end up with a single point where gn−1(x) = f(n−1)(x) is zero.
This result allows us to prove, rather easily, both of Theorems 4.3 and 4.5.
Theorem A.2 (Lagrange Interpolation Error Theorem) Let f Cn+1([a, b]) and let the nodes xk [a, b] for 0 ≤ k ≤ n. Then, for each x [a, b], there is ...