Remez inequality

In mathematics the Remez inequality, discovered by the Ukrainian mathematician E. J. Remez in 1936, 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 the "n"th degree 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

:$sup\_\{p\; in\; pi\_n(sigma)\}\; ||p||\_infty=||T\_n||\_infty$

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 $[1,\; +infty]$, hence :$||T\_n||\_infty\; =\; T\_n(1+sigma)$.

References

*cite journal|last = Remez|first = E. J.|title = Sur une propriété des polynômes de Tchebyscheff|journal = Comm. Inst. Sci. Kharkow|volume = 13|date = 1936|pages = 93–95
*cite journal|last = Bojanov|first = Borislav|title = Elementary Proof of the Remez Inequality|journal = The American Mathematical Monthly|volume = 100|issue = 5|date = May 1993|pages = 483–485|url = http://links.jstor.org/sici?sici=0002-9890%28199305%29100%3A5%3C483%3AEPOTRI%3E2.0.CO%3B2-T&size=LARGE|month = May|year = 1993|doi = 10.2307/2324304