# Remez inequality

In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez (Remez 1936), gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials.

## The inequality

Let σ be an arbitrary fixed positive number. Define the class of polynomials π$_n$(σ) to be those polynomials p of degree n for which
    $|p(x)|\leq 1$
on some set of measure ≥ 2 contained in the closed interval [−1, 1+σ]. Then the Remez inequality states that
    $\sup _{p\in \pi _{n}(\sigma )}\left\|p\right\|_{\infty }=\left\|T_{n}\right\|_{\infty }$
where T$_n$(x) is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval [−1, 1+σ].
Observe that T$_n$ is increasing on $[1,+\infty ]$, hence
    $\|T_{n}\|_{\infty }=T_{n}(1+\sigma ).$
The R.i., combined with an estimate on Chebyshev polynomials, implies the following corollary: If J ⊂ R is a finite interval, and E ⊂ J is an arbitrary measurable set, then
| $\max _{x\in J}|p(x)|\leq \left({\frac {4\,\,\operatorname {mes} J}{\operatorname {mes} E}}\right)^{n}\sup _{x\in E}|p(x)|$ | (⁎) |
for any polynomial p of degree n.

## Extensions: Nazarov–Turán lemma

Inequalities similar to (⁎) have been proved for different classes of functions, and are known as Remez-type inequalities. One important example is Nazarov's inequality for exponential sums (Nazarov 1993):
    Nazarov's inequality. Let
       $p(x)=\sum _{k=1}^{n}a_{k}e^{\lambda _{k}x}$
    be an exponential sum (with arbitrary λ$_k$ ∈C), and let J ⊂ R be a finite interval, E ⊂ J—an arbitrary measurable set. Then
       $\max _{x\in J}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\left({\frac {C\,\,\operatorname {mes} J}{\operatorname {mes} E}}\right)^{n-1}\sup _{x\in E}|p(x)|~,$
    where C > 0 is a numerical constant.
In the special case when λ$_k$ are pure imaginary and integer, and the subset E is itself an interval, the inequality was proved by Pál Turán and is known as Turán's lemma.
This inequality also extends to $L^{p}(\mathbb {T} ),\ 0\leq p\leq 2$ in the following way
    $\|p\|_{L^{p}(\mathbb {T} )}\leq e^{A(n-1)\operatorname {mes} (\mathbb {T} \setminus E)}\|p\|_{L^{p}(E)}$
for some A > 0 independent of p, E, and n. When
    $\operatorname {mes} E<1-{\frac {\log n}{n}}$
a similar inequality holds for p > 2. For p = ∞ there is an extension to multidimensional polynomials.
Proof: Applying Nazarov's lemma to $E=E_{\lambda }=\{x:|p(x)|\leq \lambda \},\ \lambda >0$ leads to
    $\max _{x\in J}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\left({\frac {C\,\,\operatorname {mes} J}{\operatorname {mes} E_{\lambda }}}\right)^{n-1}\sup _{x\in E_{\lambda }}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\left({\frac {C\,\,\operatorname {mes} J}{\operatorname {mes} E_{\lambda }}}\right)^{n-1}\lambda$
thus
    $\operatorname {mes} E_{\lambda }\leq C\,\,\operatorname {mes} J\left({\frac {\lambda e^{\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}}{\max _{x\in J}|p(x)|}}\right)^{\frac {1}{n-1}}$
Now fix a set $E$ and choose $\lambda$ such that $\operatorname {mes} E_{\lambda }\leq {\tfrac {1}{2}}\operatorname {mes} E$, that is
    $\lambda =\left({\frac {\operatorname {mes} E}{2C\operatorname {mes} J}}\right)^{n-1}e^{-\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\max _{x\in J}|p(x)|$
Note that this implies:
* $\operatorname {mes} E\setminus E_{\lambda }\geq {\tfrac {1}{2}}\operatorname {mes} E.$
* $\forall x\in E\setminus E_{\lambda }:|p(x)|>\lambda .$
Now
    ${\begin{aligned}\int _{x\in E}|p(x)|^{p}\,{\mbox{d}}x&\geq \int _{x\in E\setminus E_{\lambda }}|p(x)|^{p}\,{\mbox{d}}x\\[6pt]&\geq \lambda ^{p}{\frac {1}{2}}\operatorname {mes} E\\[6pt]&={\frac {1}{2}}\operatorname {mes} E\left({\frac {\operatorname {mes} E}{2C\operatorname {mes} J}}\right)^{p(n-1)}e^{-p\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\max _{x\in J}|p(x)|^{p}\\[6pt]&\geq {\frac {1}{2}}{\frac {\operatorname {mes} E}{\operatorname {mes} J}}\left({\frac {\operatorname {mes} E}{2C\operatorname {mes} J}}\right)^{p(n-1)}e^{-p\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\int _{x\in J}|p(x)|^{p}\,{\mbox{d}}x,\end{aligned}}$
which completes the proof.

## Pólya inequality

One of the corollaries of the Remez inequality is the Pólya inequality, which was proved by George Pólya (Pólya 1928), and states that the Lebesgue measure of a sub-level set of a polynomial p of degree n is bounded in terms of the leading coefficient LC(p) as follows:
    $\operatorname {mes} \left\{x\in \mathbb {R} :\left|P(x)\right|\leq a\right\}\leq 4\left({\frac {a}{2\mathrm {LC} (p)}}\right)^{1/n},\quad a>0~.$