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Remez inequality

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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

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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

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

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 , hence

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

()

for any polynomial p of degree n.

Extensions: Nazarov–Turán lemma

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

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 leads to

thus

Now fix a set and choose such that , that is

Note that this implies:

Now

which completes the proof.

Pólya inequality

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One of the corollaries of the Remez inequality 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:

References

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  • 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. 100 (5). Mathematical Association of America: 483–485. doi:10.2307/2324304. JSTOR 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.