---
tag: 8S129JX919
kind: statement
role: theorem
name: Remez inequality
msc: [41A17, 26D05, 26C05]
uses: [M6EAR1JHPP, 0DPF6WRZW6]
about: [M6EAR1JHPP]
sources:
  - url: https://en.wikipedia.org/wiki/Remez_inequality
same_as:
  wikidata: Q7311758
  wikipedia: Remez inequality
---

## Statement

Let $n \ge 0$ be an integer and let $\sigma > 0$.
Let $p$ be a real polynomial of [[degree|0DPF6WRZW6]] at most $n$.
Let $E \subseteq [-1, 1 + \sigma]$ be a Lebesgue measurable set of measure at least $2$ such that $\abs{p(x)} \le 1$ for all $x \in E$.
Then
$$\max_{-1 \le x \le 1 + \sigma} \abs{p(x)} \le T_n(1 + \sigma),$$
where $T_n$ is the [[Chebyshev polynomial|M6EAR1JHPP]] of the first kind of degree $n$.

## Remarks

The bound is sharp: $p = T_n$ and $E = [-1, 1]$ give equality, since $T_n$ is increasing on $[1, \infty)$.
Proof idea: compare $p$ with a rescaled Chebyshev polynomial whose set where it is at most $1$ in absolute value has measure $2$.
A count of sign changes, as in the minimal deviation property of $T_n$, shows that $p$ cannot be larger.

Example: for $n = 1$, a line with $\abs{p} \le 1$ on a set of measure $2$ in $[-1, 1 + \sigma]$ is at most $1 + \sigma = T_1(1 + \sigma)$ in absolute value on the whole interval.

By an affine change of variable, for an interval $I$ and a measurable $E \subseteq I$ of positive measure,
$$\sup_{I} \abs{p} \le T_n\left(\frac{2 \abs{I}}{\abs{E}} - 1\right) \sup_{E} \abs{p} \le \left(\frac{4 \abs{I}}{\abs{E}}\right)^n \sup_{E} \abs{p}.$$
Without a lower bound on the measure of $E$ there is no bound: $p(x) = c x$ is small on a tiny interval around $0$ for every $c$.

## Explanations

### L1 human

A polynomial of degree $n$ that is small on a large part of an interval cannot be huge on the rest of it.
The Remez inequality gives the exact limit, and the Chebyshev polynomial is the worst case.

### L2 human

The inequality is a quantitative form of the fact that a polynomial is determined by its values on any set of positive measure.
It extends Chebyshev's extremal property, which concerns the growth of $T_n$ outside $[-1, 1]$, to arbitrary measurable sets.
It is used to bound the size of sublevel sets of polynomials and has versions for several variables and for exponential sums.