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

[math]\displaystyle{ |p(x)| \le 1 }[/math]

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

[math]\displaystyle{ \sup_{p \in \pi_n(\sigma)} \left\|p\right\|_\infty = \left\|T_n\right\|_\infty }[/math]

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 [math]\displaystyle{ [1, +\infty] }[/math], hence

[math]\displaystyle{ \|T_n\|_\infty = T_n(1+\sigma). }[/math]

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

[math]\displaystyle{ \max_{x \in J} |p(x)| \leq \left( \frac{4 \,\, \textrm{mes } J}{\textrm{mes } E} \right)^n \sup_{x \in E} |p(x)| }[/math]

(⁎)

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

[math]\displaystyle{ p(x) = \sum_{k = 1}^n a_k e^{\lambda_k x} }[/math]

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

[math]\displaystyle{ \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)|~, }[/math]

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 [math]\displaystyle{ L^p(\mathbb{T}),\ 0\leq p\leq2 }[/math] in the following way

[math]\displaystyle{ \|p\|_{L^p(\mathbb{T})} \leq e^{A(n - 1) \textrm{mes }(\mathbb{T}\setminus E)}\|p\|_{L^p(E)} }[/math]

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

[math]\displaystyle{ \mathrm{mes} E \lt 1-\frac{\log n}{n} }[/math]

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

Proof: Applying Nazarov's lemma to [math]\displaystyle{ E=E_\lambda=\{x\,:\ |p(x)|\leq\lambda\},\ \lambda\gt 0 }[/math] leads to

[math]\displaystyle{ \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 }[/math]

thus

[math]\displaystyle{ \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}} }[/math]

Now fix a set [math]\displaystyle{ E }[/math] and choose [math]\displaystyle{ \lambda }[/math] such that [math]\displaystyle{ \textrm{mes} E_\lambda\leq\tfrac{1}{2}\textrm{mes} E }[/math], that is

[math]\displaystyle{ \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)| }[/math]

Note that this implies:

1. [math]\displaystyle{ \textrm{mes}E\setminus E_{\lambda}\ge \tfrac{1}{2} \textrm{mes}E . }[/math]
2. [math]\displaystyle{ \forall x \in E \setminus E_{\lambda} : |p(x)| \gt \lambda . }[/math]

Now

[math]\displaystyle{ \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}\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{align} }[/math]

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:

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

References

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