Equioscillation theorem

Remarks

The idea of the proof: if the error $e = f - p$ alternates at $n + 2$ points with full size $\norm{e}$, the de la Vallée Poussin lower bound gives $E_n(f) \ge \norm{e}$, so $p$ is best. Conversely, suppose $e$ has fewer than $n + 2$ alternation points. Then one can split $[a, b]$ into at most $n + 1$ pieces on which the extreme values of $e$ have one sign. A polynomial $q$ of degree at most $n$ with the same sign pattern then makes $\norm{f - p - t q} < \norm{e}$ for small $t > 0$. Uniqueness follows: if $p_1, p_2$ are best, then $(p_1 + p_2)/2$ is best, and at its $n + 2$ alternation points $p_1 = p_2$, so $p_1 - p_2$ has $n + 2$ zeros.

Example: for $f(x) = x^2$ on $[0, 1]$ and $n = 1$, the line $p(x) = x - 1/8$ is the best approximation. The error $x^2 - x + 1/8$ takes the values $1/8, -1/8, 1/8$ at $0, 1/2, 1$, which is $n + 2 = 3$ alternating extreme values, and $\norm{f - p} = 1/8$, so the error of best approximation is $E_1(f) = 1/8$. Another example: the monic polynomial of degree $n + 1$ of least norm on $[-1, 1]$ is $2^{-n} T_{n+1}$, where $T_{n+1}$ is the Chebyshev polynomial, because $T_{n+1}$ attains $\pm 1$ alternately at $n + 2$ points.

Going deeper

For a student

The theorem, due to Chebyshev, is the main characterization of minimax approximation by polynomials. It is the basis of the Remez algorithm, which moves a set of $n + 2$ reference points until the error equioscillates. It holds in the same form when $\mathcal{P}_n$ is replaced by any Haar subspace of dimension $n + 1$ on $[a, b]$. It also gives the extremal property of Chebyshev polynomials and many sharp constants in approximation theory.