Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies |
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Authors: | Szilárd Révész |
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Institution: | (1) Alfréd Rényi Mathematical Institute of the Hungarian Academy of Sciences Budapest Reáltanoda u. 13—15 1053 Hungary revesz@renyi.hu, HU |
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Abstract: | For a compact set K\subset R
d
with nonempty interior, the Markov constants M
n
(K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K .
It is known that for convex, symmetric bodies M
n
(K) = n
2
/r(K) , where r(K) is the ``half-width' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B
d
(\smallbf 0, 1) we do not have uniqueness, while for the unit cube -1,1]
d
the extremal polynomials are essentially unique.
September 9, 1999. Date revised: September 28, 2000. Date accepted: November 14, 2000. |
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Keywords: | , Markov inequality, Multivariate polynomials, Symmetric convex bodies, Supporting hyperplanes, AMS Classification,,,,,,41A17, 41A63, 41A10, |
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