Remez inequality
# 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 the nth degree 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

    $\max _{x\in J}|p(x)|\leq \left({\frac {4\,\,{\textrm {mes}}J}{{\textrm {mes}}E}}\right)^{n}\sup _{x\in E}|p(x)|$         (⁎)

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
       $\max _{x\in J}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\mathrm {mes} J}\left({\frac {C\,\,{\textrm {mes}}J}{{\textrm {mes}}E}}\right)^{n-1}\sup _{x\in E}|p(x)|~,$
    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

    $\|p\|_{L^{p}(\mathbb {T} )}\leq e^{A(n-1){\textrm {mes}}(\mathbb {T} \setminus E)}\|p\|_{L^{p}(E)}$

for some A>0 independent of p, E, and n. When

    $\mathrm {mes} E<1-{\frac {\log n}{n}}$

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

    $\max _{x\in J}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\mathrm {mes} J}\left({\frac {C\,\,{\textrm {mes}}J}{{\textrm {mes}}E_{\lambda }}}\right)^{n-1}\sup _{x\in E_{\lambda }}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\mathrm {mes} J}\left({\frac {C\,\,{\textrm {mes}}J}{{\textrm {mes}}E_{\lambda }}}\right)^{n-1}\lambda$

thus

    ${\textrm {mes}}E_{\lambda }\leq C\,\,{\textrm {mes}}J\left({\frac {\lambda e^{\max _{k}|\Re \lambda _{k}|\,\mathrm {mes} J}}{\max _{x\in J}|p(x)|}}\right)^{\frac {1}{n-1}}$

Now fix a set $E$ and choose $\lambda$ such that ${\textrm {mes}}E_{\lambda }\leq {\tfrac {1}{2}}{\textrm {mes}}E$, that is

    $\lambda =\left({\frac {{\textrm {mes}}E}{2C\mathrm {mes} J}}\right)^{n-1}e^{-\max _{k}|\Re \lambda _{k}|\,\mathrm {mes} J}\max _{x\in J}|p(x)|$

Note that this implies:

* ${\textrm {mes}}E\setminus E_{\lambda }\geq {\tfrac {1}{2}}{\textrm {mes}}E.$
* $\forall x\in E\setminus E_{\lambda }:|p(x)|>\lambda .$

Now

    ${\begin{aligned}\int _{x\in E}|p(x)|^{p}\,{\mbox{d}}x&\geq \int _{x\in E\setminus E_{\lambda }}|p(x)|^{p}\,{\mbox{d}}x\\[6pt]&\geq \lambda ^{p}{\frac {1}{2}}{\textrm {mes}}E\\[6pt]&={\frac {1}{2}}{\textrm {mes}}E\left({\frac {{\textrm {mes}}E}{2C\mathrm {mes} J}}\right)^{p(n-1)}e^{-p\max _{k}|\Re \lambda _{k}|\,\mathrm {mes} J}\max _{x\in J}|p(x)|^{p}\\[6pt]&\geq {\frac {1}{2}}{\frac {{\textrm {mes}}E}{{\textrm {mes}}J}}\left({\frac {{\textrm {mes}}E}{2C\mathrm {mes} J}}\right)^{p(n-1)}e^{-p\max _{k}|\Re \lambda _{k}|\,\mathrm {mes} J}\int _{x\in J}|p(x)|^{p}\,{\mbox{d}}x,\end{aligned}}$

which completes the proof.

## Pólya inequality

One of the corollaries of the R.i. 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:

    ${\textrm {mes}}\left\{x\in \mathbb {R} :\left|P(x)\right|\leq a\right\}\leq 4\left({\frac {a}{2\mathrm {LC} (p)}}\right)^{1/n},\quad a>0~.$
