Pólya 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_''n''(''σ'') to be those polynomials ''p'' of degree ''n'' for which

:$, p(x), \le 1$

on some set of measure ≥ 2 contained in the closed interval ��1, 1+''σ'' Then the Remez inequality states that

:$\sup_ \left\, p\right\, _\infty = \left\, T_n\right\, _\infty$

where ''T''_''n''(''x'') is the Chebyshev polynomial of degree ''n'', and the supremum norm is taken over the interval ��1, 1+''σ''

Observe that ''T''_''n'' is increasing on , +\infty/math>, hence
:$\, T_n\, _\infty = T_n(1+\sigma).$

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's inequality. Let
:: $p(x) = \sum_^n a_k e^$
: be an exponential sum (with arbitrary ''λ''_''k'' ∈C), and let ''J'' ⊂ R be a finite interval, ''E'' ⊂ ''J''—an arbitrary measurable set. Then
:: $\max_ , p(x), \leq e^ \left( \frac \right)^ \sup_ , p(x), ~,$
: where ''C'' > 0 is a numerical constant.

In the special case when ''λ_k'' 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 $L^p(\mathbb),\ 0\leq p\leq2$ in the following way

:$\, p\, _ \leq e^ \, p\, _$

for some ''A'' > 0 independent of ''p'', ''E'', and ''n''. When

:$\operatorname E <1-\frac$

a similar inequality holds for ''p'' > 2. For ''p'' = ∞ there is an extension to multidimensional polynomials.

Proof: Applying Nazarov's lemma to $E = E_\lambda = \,\ \lambda>0$ leads to

:$\max_ , p(x), \leq e^ \left( \frac \right)^ \sup_ , p(x), \leq e^ \left( \frac \right)^ \lambda$

thus

:$\operatorname E_\lambda \leq C \,\, \operatorname J\left(\frac \right)^$

Now fix a set $E$ and choose $\lambda$ such that $\operatorname E_\lambda\leq\tfrac\operatornameE$, that is

:$\lambda = \left(\frac\right)^e^\max_ , p(x),$

Note that this implies:
# $\operatornameE\setminus E_\ge \tfrac \operatornameE .$
# $\forall x \in E \setminus E_ : , p(x), > \lambda .$

Now

:\begin
\int_, p(x), ^p\,\mboxx &\geq \int_, p(x), ^p\,\mboxx \\ pt&\geq \lambda^p\frac\operatornameE \\ pt&= \frac\operatornameE \left(\frac\right)^e^\max_ , p(x), ^p \\ pt&\geq \frac \frac\left(\frac\right)^e^\int_ , p(x), ^p\,\mboxx,
\end

which completes the proof.

Pólya inequality

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:

:$\operatorname \left\ \leq 4 \left(\frac\right)^, \quad a>0~.$

References

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*{{cite journal, last = Pólya, first = G., author-link=George Pólya, title = Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete, journal = Sitzungsberichte Akad. Berlin, year = 1928, pages = 280–282

Theorems in mathematical analysis
Inequalities (mathematics)