In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez, gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials.
Let σ be an arbitrary fixed positive number. Define the class of polynomials πn(σ) to be those polynomials p of degree n for which
|p(x)|\le1
on some set of measure ≥ 2 contained in the closed interval [−1, 1+''σ'']. Then the Remez inequality states that
| \sup | |
| p\in\pin(\sigma) |
\left\|p\right\|infty=\left\|Tn\right\|infty
where Tn(x) is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval [−1, 1+''σ''].
Observe that Tn is increasing on
[1,+infty]
\|Tn\|infty=Tn(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, thenfor any polynomial p of degree n.
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's inequality. Let
p(x)=
| n | |
| \sum | |
| k=1 |
ak
| λkx | |
| e |
be an exponential sum (with arbitrary λk ∈C), and let J ⊂ R be a finite interval, E ⊂ J—an arbitrary measurable set. Then
maxx|p(x)|\leq
| maxk|\Reλk|\operatorname{mes | |
| e |
J}\left(
| C\operatorname{mes | |
| J}{\operatorname{mes}E} |
\right)n-1\supx|p(x)|~,
where C > 0 is a numerical constant.
In the special case when λk 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
Lp(T), 0\leqp\leq2
| \|p\| | |
| Lp(T) |
\leqeA(n-1)(T\setminusE)}
| \|p\| | |
| Lp(E) |
for some A > 0 independent of p, E, and n. When
\operatorname{mes}E<1-
| logn | |
| n |
a similar inequality holds for p > 2. For p = ∞ there is an extension to multidimensional polynomials.
Proof: Applying Nazarov's lemma to
E=Eλ=\{x:|p(x)|\leqλ\}, λ>0
maxx|p(x)|\leq
| maxk|\Reλk|\operatorname{mes | |
| e |
J}\left(
| C\operatorname{mes | |
| J}{\operatorname{mes} |
Eλ}\right)n-1
| \sup | |
| x\inEλ |
|p(x)|\leq
| maxk|\Reλk|\operatorname{mes | |
| e |
J}\left(
| C\operatorname{mes | |
| J}{\operatorname{mes} |
Eλ}\right)n-1λ
thus
\operatorname{mes}Eλ\leqC\operatorname{mes}J\left(
| |||||||
Now fix a set
E
λ
\operatorname{mes}Eλ\leq\tfrac{1}{2}\operatorname{mes}E
λ=\left(
| \operatorname{mes | |
| E}{2C |
\operatorname{mes}J}\right)n-1
| -maxk|\Reλk|\operatorname{mes | |
| e |
J}maxx|p(x)|
Note that this implies:
\operatorname{mes}E\setminusEλ\ge\tfrac{1}{2}\operatorname{mes}E.
\forallx\inE\setminusEλ:|p(x)|>λ.
Now
\begin{align} \intx\in|p(x)|pdx&\geq
| \int | |
| x\inE\setminusEλ |
|p(x)|pdx\\[6pt] &\geq
| ||||
| λ |
\operatorname{mes}E\\[6pt] &=
| 1 | |
| 2 |
\operatorname{mes}E\left(
| \operatorname{mes | |
| E}{2C |
\operatorname{mes}J}\right)p(n-1)
| -pmaxk|\Reλk|\operatorname{mes | |
| e |
J}maxx|p(x)|p\\[6pt] &\geq
| 1 | |
| 2 |
| \operatorname{mes | |||
|
\operatorname{mes}J}\right)p(n-1)
| -pmaxk|\Reλk|\operatorname{mes | |
| e |
J}\intx|p(x)|pdx, \end{align}
which completes the proof.
One of the corollaries of the Remez inequality is the Pólya inequality, which was proved by George Pólya, 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\R:\left|P(x)\right|\leqa\right\}\leq4\left(
| a | |
| 2LC(p) |
\right)1/n, a>0~.