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


|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 Tn(x) is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval [−1, 1+σ].

[edit] References