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On Multivariate Polynomials with Largest Gradients on Convex Bodies

✍ Scribed by András Kroó

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
106 KB
Volume
253
Category
Article
ISSN
0022-247X

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✦ Synopsis

In this paper we study the extremal polynomials for the Markov inequality on a convex symmetric body K ; ‫ޒ‬ m , that is, norm 1 polynomials on K whose gradients attain the largest value on K. It is shown that any such polynomial must coincide with the Chebyshev polynomial along a certain line. Moreover, this fact is applied to the study of uniqueness of extremal polynomials. ᮊ 2001 Academic Press Let P m , n, m g ‫ޚ‬ q , be the space of real polynomials of m variables and n total degree at most n, and consider a convex body K ; ‫ޒ‬ m symmetric about 0 g K. As usual, m Ѩ p n ² : grad p x [ , D p x [ grad p x , y n

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