In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez , gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials.
Let ÃÂ be an arbitrary fixed positive number. Define the class of polynomials ÃÂ<sub>n</sub>(ÃÂ) to be those polynomials p of degree n for which
on some set of measure âÂÂ¥ 2 contained in the closed interval [âÂÂ1, 1+ÃÂ]. Then the Remez inequality states that
where T<sub>n</sub>(x) is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval [âÂÂ1, 1+ÃÂ].
Observe that T<sub>n</sub> is increasing on , hence
The R.i., combined with an estimate on Chebyshev polynomials, implies the following corollary: If J â R is a finite interval, and E â J is an arbitrary measurable set, then
for any polynomial p of degree n.
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 :
In the special case when û<sub>k</sub> 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 = â 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.
One of the corollaries of the Remez inequality is the Pólya inequality, which was proved by George Pólya , 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: