Approximation theory/Proofs

Proof

Approximation theory

Proof that an N^th degree polynomial that gives rise to an error function that has N+2 maxima, of alternating signs and equal magnitutes, is optimal.

This is most easily seen with a graph. Let N=4 in the example. Suppose P(x) is an N^th degree polynomial that is optimal, and that P(x)-f(x) oscillates among N+2 maxima.