Remez 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

:$|p(x)|\; \backslash le\; 1$

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

:$\backslash sup\_\{p\; \backslash in\; \backslash pi\_n(\backslash sigma)\}\; \backslash left\backslash |p\backslash right\backslash |\_\backslash infty\; =\; \backslash left\backslash |T\_n\backslash right\backslash |\_\backslash infty$

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 $[1,\; +\backslash infty]$, hence

:$\backslash |T\_n\backslash |\_\backslash infty\; =\; T\_n(1+\backslash sigma).$

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

{{NumBlk|:|$\backslash max\_\{x\; \backslash in\; J\}\; |p(x)|\; \backslash leq\; \backslash left(\; \backslash frac\{4\; \backslash ,\backslash ,\; \backslash operatorname\{mes\}J\}\{\backslash operatorname\{mes\}E\}\; \backslash right)^n\; \backslash sup\_\{x\; \backslash in\; E\}\; |p(x)|$|{{EquationRef|⁎}}}}

for any polynomial p of degree n.

Inequalities similar to ({{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 {{harv|Nazarov|1993}}:

: Nazarov's inequality. Let

:: $p(x)\; =\; \backslash sum\_\{k=1\}^n\; a\_k\; e^\{\backslash lambda\_k\; x\}$

: be an exponential sum (with arbitrary λ_k ∈C), and let J ⊂ R be a finite interval, E ⊂ J—an arbitrary measurable set. Then

:: $\backslash max\_\{x\; \backslash in\; J\}\; |p(x)|\; \backslash leq\; e^\{\backslash max\_k\; |\backslash Re\; \backslash lambda\_k|\; \backslash ,\; \backslash operatorname\{mes\}J\}\; \backslash left(\; \backslash frac\{C\; \backslash ,\backslash ,\; \backslash operatorname\{mes\}J\}\{\backslash operatorname\{mes\}E\}\; \backslash right)^\{n-1\}\; \backslash sup\_\{x\; \backslash in\; E\}\; |p(x)|~,$

: 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 $L^p(\backslash mathbb\{T\}),\backslash \; 0\backslash leq\; p\backslash leq2$ in the following way

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

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

:

a similar inequality holds for p > 2. For p = ∞ there is an extension to multidimensional polynomials.

Proof: Applying Nazarov's lemma to $E\; =\; E\_\backslash lambda\; =\; \backslash \{x\; :\; |p(x)|\backslash leq\backslash lambda\backslash \},\backslash \; \backslash lambda>0$ leads to

:$\backslash max\_\{x\; \backslash in\; J\}\; |p(x)|\; \backslash leq\; e^\{\backslash max\_k\; |\backslash Re\; \backslash lambda\_k|\; \backslash ,\; \backslash operatorname\{mes\}\; J\}\; \backslash left(\; \backslash frac\{C\; \backslash ,\backslash ,\; \backslash operatorname\{mes\}\; J\}\{\backslash operatorname\{mes\}\; E\_\backslash lambda\}\; \backslash right)^\{n-1\}\; \backslash sup\_\{x\; \backslash in\; E\_\backslash lambda\}\; |p(x)|\; \backslash leq\; e^\{\backslash max\_k\; |\backslash Re\; \backslash lambda\_k|\; \backslash ,\; \backslash operatorname\{mes\}\; J\}\; \backslash left(\; \backslash frac\{C\; \backslash ,\backslash ,\; \backslash operatorname\{mes\}\; J\}\{\backslash operatorname\{mes\}\; E\_\backslash lambda\}\; \backslash right)^\{n-1\}\; \backslash lambda$

thus

:$\backslash operatorname\{mes\}\; E\_\backslash lambda\; \backslash leq\; C\; \backslash ,\backslash ,\; \backslash operatorname\{mes\}\; J\backslash left(\backslash frac\{\backslash lambda\; e^\{\backslash max\_k\; |\backslash Re\; \backslash lambda\_k|\; \backslash ,\; \backslash operatorname\{mes\}J\}\}\{\backslash max\_\{x\; \backslash in\; J\}\; |p(x)|\}\; \backslash right)^\{\backslash frac\{1\}\{n-1\}\}$

Now fix a set $E$ and choose $\backslash lambda$ such that $\backslash operatorname\{mes\}\; E\_\backslash lambda\backslash leq\backslash tfrac\{1\}\{2\}\backslash operatorname\{mes\}E$, that is

:$\backslash lambda\; =\; \backslash left(\backslash frac\{\backslash operatorname\{mes\}\; E\}\{2C\; \backslash operatorname\{mes\}J\}\backslash right)^\{n-1\}e^\{-\backslash max\_k\; |\backslash Re\; \backslash lambda\_k|\; \backslash ,\; \backslash operatorname\{mes\}J\}\backslash max\_\{x\; \backslash in\; J\}\; |p(x)|$

Note that this implies:

1. $\backslash operatorname\{mes\}E\backslash setminus\; E\_\{\backslash lambda\}\backslash ge\; \backslash tfrac\{1\}\{2\}\; \backslash operatorname\{mes\}E\; .$
2. $\backslash forall\; x\; \backslash in\; E\; \backslash setminus\; E\_\{\backslash lambda\}\; :\; |p(x)|\; >\; \backslash lambda\; .$

Now

:$\backslash begin\{align\}$

\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{align}

which completes the proof.

One of the corollaries of the Remez inequality is the Pólya inequality, which was proved by George Pólya {{harv|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:

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

• {{cite journal|last = Remez|first = E. J.|author-link=Evgeny Yakovlevich Remez|title = Sur une propriété des polynômes de Tchebyscheff|journal = Comm. Inst. Sci. Kharkow|volume = 13|year = 1936|pages = 93–95}}
• {{cite journal|last = Bojanov|first = B.|title = Elementary Proof of the Remez Inequality|journal = The American Mathematical Monthly|volume = 100|issue = 5|date = May 1993|pages = 483–485|doi = 10.2307/2324304|jstor = 2324304|publisher = Mathematical Association of America}}
• {{cite journal|last = Fontes-Merz|first = N.|title = A multidimensional version of Turan's lemma|journal = Journal of Approximation Theory|volume = 140 |issue = 1|pages = 27–30|year=2006|doi = 10.1016/j.jat.2005.11.012|doi-access = free}}
• {{cite journal|last = Nazarov|first = F.|author-link=Fedor Nazarov|title = Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type|journal = Algebra i Analiz|volume = 5|issue = 4|pages = 3–66|year=1993}}
• {{cite book|last = Nazarov|first = F.| chapter=Complete Version of Turan's Lemma for Trigonometric Polynomials on the Unit Circumference |author-link=Fedor Nazarov|title = Complex Analysis, Operators, and Related Topics|volume = 113|pages =239–246|year=2000|doi = 10.1007/978-3-0348-8378-8_20|isbn = 978-3-0348-9541-5}}
• {{cite journal|last = Pólya|first = G.|author-link=George Pólya| title = Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete|journal = Sitzungsberichte Akad. Berlin|year = 1928|pages = 280–282}}

Category:Theorems in mathematical analysis

Category:Inequalities (mathematics)