Talk:Remez inequality

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[edit] Is this the right inequality?

I've done a bit of digging around to try to nail down the exact form in which this inequality was originally given, without much success. I don't have access to the JSTOR articles, for example.

I did find this useful web page, constructed by professor Tamás Erdélyi, who has written several papers about Remez-type inequalities. In The Remez Inequality for Linear Combinations of Shifted Gaussians (page 3) Erdélyi says

"The classical Remez inequality states that if p is a polynomial of degree at most n, s ∈ (0, 2), and
m(\lbrace x \in [-1, 1] : |p(x)| \le 1 \rbrace) \ge 2-s,
then
\lVert p(x) \rVert_{[-1,1]} \le T_n \left( \frac{2+s}{2-s} \right),
where Tn(x) = cos(n arccos x) is the Chebyshev polynomial of degree n."

In other words, the supremum of |p(x)| on the interval [−1, 1] is bounded by a value of Tn(y), y on the interval (1, ∞), the exact point y in that interval depending on the measure of a set within which |p(x)| ≤ 1.

I can't square this up with the way the inequality is stated in this article, so I'm confused. Can anybody clear this up for me? DavidCBryant 22:29, 20 July 2007 (UTC)