Remez inequality


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.

The inequality

Let σ be an arbitrary fixed positive number. Define the class of polynomials πn to be those polynomials p of the nth degree for which
on some set of measure ≥ 2 contained in the closed interval . Then the Remez inequality states that
where Tn is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval .
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 :
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:
  1. .
  2. .
Now
which completes the proof.

Pólya inequality

One of the corollaries of the R.i. 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 as follows: