Bernstein classes
Annales de l'Institut Fourier, Volume 47 (1997) no. 3, pp. 825-858.

One of the classical Bernstein inequalities compares the maxima of a polynomial of a given degree on the interval [-1,1] and on the ellipse in the complex plane with the focuses -1, 1 and the semiaxes R. We prove a similar inequality for a branch of an algebraic function of a given degree on the maximal disk of its regularity, with the explicitly given constant, depending on the degree only. In particular, this improves a recent inequality of Fefferman and Narasimhan and answers one of their questions. We present in detail various properties of the classes of functions, satisfying Bernstein type inequalities and various approaches to establishing such inequalities.

Une des inégalités classiques de Bernstein compare les maximas d’un polynôme de degré donné sur l’intervalle [-1,1] et sur l’ellipse du plan complexe de foyers -1, 1 et de semi-axes R. Nous démontrons une inégalité similaire pour une branche de fonction algébrique de degré donné sur son disque maximal de régularité avec une constante donnée explicitement, dépendant seulement du degré. En particulier cela améliore une inégalité récente due à Fefferman et Narasimhan et cela répond à une de leurs questions. Nous présentons en détail diverses propriétés de classes de fonctions satisfaisant des inégalités du type de celle de Bernstein et diverses approches pour établir ces inégalités.

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Roytwarf, N.; Yomdin, Yosef. Bernstein classes. Annales de l'Institut Fourier, Volume 47 (1997) no. 3, pp. 825-858. doi : 10.5802/aif.1582. http://www.numdam.org/articles/10.5802/aif.1582/

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