Inmathematics , theRemez inequality , discovered by the Soviet mathematicianEvgeny Yakovlevich Remez (Remez 1936 ), gives abound on thesup norms of certainpolynomials , the bound being attained by theChebyshev polynomials .
Letσ be an arbitrary fixed positive number. Define the class of polynomials πn σ ) to be those polynomialsp ofdegree n for which
| p ( x ) | ≤ 1 {\displaystyle |p(x)|\leq 1} on some set ofmeasure ≥ 2 contained in theclosed interval [−1, 1+σ ]. Then theRemez inequality states that
sup p ∈ π n ( σ ) ‖ p ‖ ∞ = ‖ T n ‖ ∞ {\displaystyle \sup _{p\in \pi _{n}(\sigma )}\left\|p\right\|_{\infty }=\left\|T_{n}\right\|_{\infty }} whereT n x ) is theChebyshev polynomial of degreen , and thesupremum norm is taken over the interval [−1, 1+σ ].
Observe thatT n [ 1 , + ∞ ] {\displaystyle [1,+\infty ]}
‖ T n ‖ ∞ = T n ( 1 + σ ) . {\displaystyle \|T_{n}\|_{\infty }=T_{n}(1+\sigma ).} The R.i., combined with an estimate on Chebyshev polynomials, implies the followingcorollary : IfJ ⊂ R is a finite interval, andE ⊂ J is an arbitrarymeasurable set , then
for any polynomialp of degreen .
[ edit ] Inequalities similar to (⁎ proved for different classes offunctions , and are known as Remez-type inequalities. One important example isNazarov 's inequality forexponential sums (Nazarov 1993 ):
Nazarov's inequality . Letp ( x ) = ∑ k = 1 n a k e λ k x {\displaystyle p(x)=\sum _{k=1}^{n}a_{k}e^{\lambda _{k}x}} be an exponential sum (with arbitraryλ k C ), and letJ ⊂ R be a finite interval,E ⊂ J —an arbitrary measurable set. Thenmax x ∈ J | p ( x ) | ≤ e max k | ℜ λ k | mes J ( C mes J mes E ) n − 1 sup x ∈ E | p ( x ) | , {\displaystyle \max _{x\in J}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\left({\frac {C\,\,\operatorname {mes} J}{\operatorname {mes} E}}\right)^{n-1}\sup _{x\in E}|p(x)|~,} whereC > 0 is a numerical constant. In the special case whenλk are pure imaginary and integer, and the subsetE is itself an interval, the inequality was proved byPál Turán and is known as Turán's lemma.
This inequality also extends toL p ( T ) , 0 ≤ p ≤ 2 {\displaystyle L^{p}(\mathbb {T} ),\ 0\leq p\leq 2}
‖ p ‖ L p ( T ) ≤ e A ( n − 1 ) mes ( T ∖ E ) ‖ p ‖ L p ( E ) {\displaystyle \|p\|_{L^{p}(\mathbb {T} )}\leq e^{A(n-1)\operatorname {mes} (\mathbb {T} \setminus E)}\|p\|_{L^{p}(E)}} for someA > 0 independent ofp ,E , andn . When
mes E < 1 − log n n {\displaystyle \operatorname {mes} E<1-{\frac {\log n}{n}}} a similar inequality holds forp > 2. Forp = ∞ there is an extension to multidimensional polynomials.
Proof: Applying Nazarov's lemma toE = E λ = { x : | p ( x ) | ≤ λ } , λ > 0 {\displaystyle E=E_{\lambda }=\{x:|p(x)|\leq \lambda \},\ \lambda >0}
max x ∈ J | p ( x ) | ≤ e max k | ℜ λ k | mes J ( C mes J mes E λ ) n − 1 sup x ∈ E λ | p ( x ) | ≤ e max k | ℜ λ k | mes J ( C mes J mes E λ ) n − 1 λ {\displaystyle \max _{x\in J}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\left({\frac {C\,\,\operatorname {mes} J}{\operatorname {mes} E_{\lambda }}}\right)^{n-1}\sup _{x\in E_{\lambda }}|p(x)|\leq e^{\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\left({\frac {C\,\,\operatorname {mes} J}{\operatorname {mes} E_{\lambda }}}\right)^{n-1}\lambda } thus
mes E λ ≤ C mes J ( λ e max k | ℜ λ k | mes J max x ∈ J | p ( x ) | ) 1 n − 1 {\displaystyle \operatorname {mes} E_{\lambda }\leq C\,\,\operatorname {mes} J\left({\frac {\lambda e^{\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}}{\max _{x\in J}|p(x)|}}\right)^{\frac {1}{n-1}}} Now fix a setE {\displaystyle E} λ {\displaystyle \lambda } mes E λ ≤ 1 2 mes E {\displaystyle \operatorname {mes} E_{\lambda }\leq {\tfrac {1}{2}}\operatorname {mes} E}
λ = ( mes E 2 C mes J ) n − 1 e − max k | ℜ λ k | mes J max x ∈ J | p ( x ) | {\displaystyle \lambda =\left({\frac {\operatorname {mes} E}{2C\operatorname {mes} J}}\right)^{n-1}e^{-\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\max _{x\in J}|p(x)|} Note that this implies:
mes E ∖ E λ ≥ 1 2 mes E . {\displaystyle \operatorname {mes} E\setminus E_{\lambda }\geq {\tfrac {1}{2}}\operatorname {mes} E.} ∀ x ∈ E ∖ E λ : | p ( x ) | > λ . {\displaystyle \forall x\in E\setminus E_{\lambda }:|p(x)|>\lambda .} Now
∫ x ∈ E | p ( x ) | p d x ≥ ∫ x ∈ E ∖ E λ | p ( x ) | p d x ≥ λ p 1 2 mes E = 1 2 mes E ( mes E 2 C mes J ) p ( n − 1 ) e − p max k | ℜ λ k | mes J max x ∈ J | p ( x ) | p ≥ 1 2 mes E mes J ( mes E 2 C mes J ) p ( n − 1 ) e − p max k | ℜ λ k | mes J ∫ x ∈ J | p ( x ) | p d x , {\displaystyle {\begin{aligned}\int _{x\in E}|p(x)|^{p}\,{\mbox{d}}x&\geq \int _{x\in E\setminus E_{\lambda }}|p(x)|^{p}\,{\mbox{d}}x\\[6pt]&\geq \lambda ^{p}{\frac {1}{2}}\operatorname {mes} E\\[6pt]&={\frac {1}{2}}\operatorname {mes} E\left({\frac {\operatorname {mes} E}{2C\operatorname {mes} J}}\right)^{p(n-1)}e^{-p\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\max _{x\in J}|p(x)|^{p}\\[6pt]&\geq {\frac {1}{2}}{\frac {\operatorname {mes} E}{\operatorname {mes} J}}\left({\frac {\operatorname {mes} E}{2C\operatorname {mes} J}}\right)^{p(n-1)}e^{-p\max _{k}|\Re \lambda _{k}|\,\operatorname {mes} J}\int _{x\in J}|p(x)|^{p}\,{\mbox{d}}x,\end{aligned}}} which completes the proof.
One of the corollaries of the Remez inequality is thePólya inequality , which was proved byGeorge Pólya (Pólya 1928 ), and states that theLebesgue measure of a sub-level set of a polynomialp of degreen is bounded in terms of the leadingcoefficient LC(p ) as follows:
mes { x ∈ R : | P ( x ) | ≤ a } ≤ 4 ( a 2 L C ( p ) ) 1 / n , a > 0 . {\displaystyle \operatorname {mes} \left\{x\in \mathbb {R} :\left|P(x)\right|\leq a\right\}\leq 4\left({\frac {a}{2\mathrm {LC} (p)}}\right)^{1/n},\quad a>0~.} 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 .100 (5). Mathematical Association of America: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.doi :10.1016/j.jat.2005.11.012 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 . Vol. 113. pp. 239– 246.doi :10.1007/978-3-0348-8378-8_20 .ISBN 978-3-0348-9541-5 Pólya, G. (1928). "Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete".Sitzungsberichte Akad. Berlin :280– 282.
