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 π("σ") to be those polynomials "p" of the "n"th degree for which on some set of measure ≥ 2 contained in the closed interval [−1, 1+"σ"]. Then the Remez inequality states that where "T"("x") is the Chebyshev polynomial of degree "n", and the supremum norm is taken over the interval [−1, 1+"σ ]. Observe that T" is increasing on formula_3, 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 "λ" 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 formula_7 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 formula_10 leads to thus Now fix a set formula_13 and choose formula_14 such that formula_15, that is Note that this implies: 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("p") as follows: