In the book Proofs from The Book by Aigner and Ziegler there is a proof of 'Chebyshev's Theorem' which states that if $p(x)$ is a real polynomial of degree n with leading coefficient $1$ then
$$ \max_{-1 \leq x \leq 1} |p(x)| \geq \frac{1}{2^{n-1}}$$
In the proof $g( \theta ) = p( \cos \theta )$ is written as a cosine polynomial $ \sum_{k=0}^n \lambda_k \cos(k \theta)$ and it is then proved that this can't be less than $|\lambda_n|$ (which happens to be $\frac{1}{2^{n-1}}$) everywhere.
Afterwards it is claimed that 'The reader can easily complete the analysis' to show that the Chebyshev polynomials are the only ones for which equality occurs in the above inequality. I haven't been able to figure this out. Can someone explain why this is true? (preferably building on the proof as it was done in the book or by some other simple argument.)
edit: here by Chebyshev polynomial I actually mean the monic polynomial that you get after dividing the $n$-th Chebyshev polynomial by $2^{n-1}$.
Best Answer
Let $p(x)$ be a monic polynomial of degree $n$. Let $T_n(x)$ be the Chebyshev polynomial of degree $n$. Assume that $|p(x)|\leq 1/2^{n-1}$ for all $x\in[-1,1]$.
Set $q(x)=T_n(x)-p(x)$. Then since $T_n$ and $p$ are both monic polynomials of degree $n$, $q$ is a polynomial of degree at most $n-1$.
Now let $x_k=\cos(k\pi/n)$ for $k=0, 1, \ldots, n$. At these points $T_n(x_k)=(-1)^k/2^{n-1}$ by $T_n$'s definition. Now $$q(x_k)=T_n(x_k)-p(x_k)=(-1)^k/2^{n-1}-p(x_k)$$
But $|p(x_k)|\leq 1/2^{n-1}$ for each $k=0, 1,\ldots, n$. So that $q(x_k)\leq 0$ when $k$ is odd and $q(x_k)\geq 0$ when $k$ is even.
The Intermediate Value theorem implies that for each $k=0, 1,\ldots, n-1$ the polynomial $q(x)$ has at least one zero between $x_k$ and $x_{k+1}$. Thus $q(x)$ has at least $n$ zeros in the interval $[-1,1]$. But the degree of $q$ is less than $n$. Thus $q$ is equivalently $0$. Thus $T_n=p$.