Remez inequality

In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez Template:Harv, 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

$|p(x)|\le 1$

on some set of measure ≥ 2 contained in the closed interval [−1, 1+σ]. Then the Remez inequality states that

$\underset{p\in {\pi }_n(\sigma )}{\sup }{\Vert p\Vert }_{\mathrm{\infty }}={\Vert T_n\Vert }_{\mathrm{\infty }}$

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 $[1,+\mathrm{\infty }]$, hence

$\Vert T_n{\Vert }_{\mathrm{\infty }}=T_n(1+\sigma ).$

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 Template:NumBlk for any polynomial p of degree n.

Extensions: Nazarov–Turán lemma

Inequalities similar to (Template:EquationNote) 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 Template:Harv:

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

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

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 $L^p(𝕋),0\le p\le 2$ in the following way

$\Vert p{\Vert }_{L^p(𝕋)}\le e^{A(n-1)\mathrm{mes}(𝕋\setminus E)}\Vert p{\Vert }_{L^p(E)}$

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 $E=E_{\lambda }=\{x:|p(x)|\le \lambda \},\lambda >0$ leads to

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

thus

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

Now fix a set $E$ and choose $\lambda$ such that $\mathrm{mes}{E}_{\lambda }\le \frac{1}{2}\mathrm{mes}E$, that is

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

Note that this implies:

1. $\mathrm{mes}E\setminus E_{\lambda }\ge \frac{1}{2}\mathrm{mes}E.$
2. $\mathrm{\forall }x\in E\setminus E_{\lambda }:|p(x)|>\lambda .$

Now

$\begin{array}{cc}\underset{x\in E}{\int }|p(x){|}^p\text{d}x& \ge \underset{x\in E\setminus E_{\lambda }}{\int }|p(x){|}^p\text{d}x\\ & \ge {\lambda }^p\frac{1}{2}\mathrm{mes}E\\ & =\frac{1}{2}\mathrm{mes}E{\left(\frac{\mathrm{mes}E}{2C\mathrm{mes}J}\right)}^{p(n-1)}{e}^{-p\underset{k}{\max }|\mathrm{\Re }{\lambda }_k|\mathrm{mes}J}\underset{x\in J}{\max }|p(x){|}^p\\ & \ge \frac{1}{2}\frac{\mathrm{mes}E}{\mathrm{mes}J}{\left(\frac{\mathrm{mes}E}{2C\mathrm{mes}J}\right)}^{p(n-1)}{e}^{-p\underset{k}{\max }|\mathrm{\Re }{\lambda }_k|\mathrm{mes}J}\underset{x\in J}{\int }|p(x){|}^p\text{d}x,\end{array}$

which completes the proof.

Pólya inequality

One of the corollaries of the Remez inequality is the Pólya inequality, which was proved by George Pólya Template:Harv, 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:

$\mathrm{mes}\{x\in ℝ:|P(x)|\le a\}\le 4{\left(\frac{a}{2\mathrm{L}\mathrm{C}(p)}\right)}^{1/n},a>0.$

References

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