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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.
"The classical Remez inequality states that if p is a polynomial of degree at most n, s ∈ (0, 2), and
then
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? DavidCBryant22:29, 20 July 2007 (UTC)[reply]
Hi,
both versions are equivalent. Indeed, let be a linear function that maps onto . If is a polynomial on such that
,
then satisfies
,
and vice versa. Of course, (with some abuse of notation, since the first norm is on , whereas the second one is on .)
The inequality in the Wiki-article states that
,
whereas Erdélyi's version is
.
To make the (equivalent) assumptions (1), (2) identical to those of Erdélyi, you should take
,
and then the conclusions (3), (4) are also identical.