In mathematics, the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attributed to Chebyshev.

Let ${\displaystyle f}$ be a continuous function from ${\displaystyle [a,b]}$ to ${\displaystyle \mathbb {R} }$. Among all the polynomials of degree ${\displaystyle \leq n}$, the polynomial ${\displaystyle g}$ minimizes the uniform norm of the difference ${\displaystyle \|f-g\|_{\infty }}$ if and only if there are ${\displaystyle n+2}$ points ${\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n+1}\leq b}$ such that ${\displaystyle f(x_{i})-g(x_{i})=\sigma (-1)^{i}\|f-g\|_{\infty }}$ where ${\displaystyle \sigma }$ is either -1 or +1.

That is, the polynomial ${\displaystyle g}$ oscillates above and below ${\displaystyle f}$ at the interpolation points, and does so to the same degree.

Let us define the equioscillation condition as the condition in the theorem statement, that is, the condition that there exists ${\displaystyle n+2}$ ordered points in the interval such that the difference ${\displaystyle f(x_{i})-g(x_{i})}$ alternates in sign and is equal in magnitude to the uniform-norm of ${\displaystyle f(x)-g(x)}$.

We need to prove that this condition is 'sufficient' for the polynomial ${\displaystyle g}$ being the best uniform approximation to ${\displaystyle f}$, and we need to prove that this condition is 'necessary' for a polynomial to be the best uniform approximation.

Assume by contradiction that a polynomial ${\displaystyle p(x)}$ of degree less than or equal to ${\displaystyle n}$ existed that provides a uniformly better approximation to ${\displaystyle f}$, which means that ${\displaystyle \|f-p\|_{\infty }<\|f-g\|_{\infty }}$. Then the polynomial

is also of degree less than or equal to ${\displaystyle n}$. However, for every ${\displaystyle x_{i}}$ of the ${\displaystyle n+2}$ points ${\displaystyle x_{0},x_{1},...x_{n}}$, we know that ${\displaystyle |p(x_{i})-f(x_{i})|<|g(x)-f(x)|}$ because ${\displaystyle |p(x_{i})-f(x_{i})|\leq \|f-p\|_{\infty }}$ and ${\displaystyle \|f-p||_{\infty }<\|f-g\|_{\infty }}$ (since ${\displaystyle p}$ is a better approximation than ${\displaystyle g}$).

Therefore, ${\displaystyle h(x_{i})=(g(x_{i})-f(x_{i}))-(p(x_{i})-f(x_{i}))}$ will have the same sign as ${\displaystyle g(x_{i})-f(x_{i})}$ (because the second term has a smaller magnitude than the first). Thus, ${\displaystyle h(x_{i})}$ will also alternate sign on these ${\displaystyle n+2}$ points, and thus have at least ${\displaystyle n+1}$ roots. However, since ${\displaystyle h}$ is a 'polynomial' of at most degree ${\displaystyle n}$, it should only have at most ${\displaystyle n}$ roots. This is a contradiction.