 Equioscillation theorem

Let $f$ be a real continuous function on a nondegenerate compact interval $[a,b]$, and let $p$ be a real polynomial of degree at most $m$. Then $p$ is a best uniform approximation from the polynomials of degree at most $m$ if and only if there are $m+2$ ordered pointswith $\sigma \in \{-1,1\}$. The best polynomial is unique. The real scalar field and ordered interval are part of this alternation statement.

For sufficiency with nonzero error, a strictly better polynomial $q$ would make $q-p$ have alternating strict signs at these points, forcing at least $m+1$ distinct zeros by the intermediate value theorem. A nonzero degree-at-most-$m$ polynomial cannot have so many zeros. Necessity follows from the Kolmogorov criterion for uniform approximation and the zero-counting property of the polynomial Chebyshev system. If the error is zero, optimality is immediate.

The Equioscillation theorem, also known as the Weierstrass Approximation Theorem, is primarily associated with the field of approximation theory, particularly in the context of polynomial approximation of continuous functions. It is most commonly framed in the setting of the uniform approximation of continuous functions on closed intervals.