Remez inequality
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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.
[edit] The inequality
Let σ be an arbitrary fixed positive number. Define the class of polynomials πn(σ) to be those polynomials p of the nth degree for which
on some set of measure ≥ 2 contained in the closed interval [−1, 1+σ]. Then the Remez inequality states that
where Tn(x) is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval [−1, 1+σ].
[edit] References
- Remez, E. J. (1936). "Sur une propriété des polynômes de Tchebyscheff". Comm. Inst. Sci. Kharkow 13: 93–95.
- Bojanov, Borislav (May 1993). "Elementary Proof of the Remez Inequality". The American Mathematical Monthly 100 (5): 483–485. doi: .