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.
Let σ be an arbitrary fixed positive number. Define the class of polynomials πn(σ) to be those polynomials p of the nth degree 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
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
Now fix a set and choose such that , that is
Note that this implies:
which completes the proof.
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:
- 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. JSTOR 2324304.
- Fontes-Merz, N. (2006). "A multidimensional version of Turan's lemma". Journal of Approximation Theory. 140 (1): 27–30.
- 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. pp. 239–246.
- Pólya, G. (1928). "Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete". Sitzungsberichte Akad. Berlin: 280–282.