A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials

Szilárd Gy. Révész¹

¹ A. Rényi Institute of Mathematics Hungarian Academy of Sciences P.O.B. 127 Budapest, 1364 Hungary

Annales Polonici Mathematici, Tome 88 (2006) no. 3, pp. 229-245.

Résumé

We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in a domain where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses in a convex domain $K$. The other, pluripotential-theoretic approach, mainly due to Baran, works for even more general sets, and uses the pluricomplex Green function (the Zaharjuta–Siciak extremal function). When the inscribed ellipse method is applied on nonsymmetric convex domains, a key role is played by the generalized Minkowski functional $\alpha(K,x)$. With the aid of this functional, our current knowledge of the best constant in the multivariate Berstein inequality is precise within a constant $\sqrt{2}$ factor. Recently L. Milev and the author derived the exact yield of the inscribed ellipse method in the case of the simplex, and a number of numerical improvements were obtained compared to the general estimates known. Here we compare the yields of this real, geometric method and the results of the complex, pluripotential-theoretical approach in the case of the simplex. We observe a few remarkable facts, comment on the existing conjectures, and formulate a number of new hypotheses.

Détail

DOI : 10.4064/ap88-3-3

Keywords: compare yields methods obtain bernstein type pointwise estimates derivative multivariate polynomial domain where polynomial assumed have sup norm method due sarantopoulos relies inscribing ellipses convex domain other pluripotential theoretic approach mainly due baran works even general sets uses pluricomplex green function zaharjuta siciak extremal function inscribed ellipse method applied nonsymmetric convex domains key role played generalized minkowski functional alpha aid functional current knowledge best constant multivariate berstein inequality precise within constant sqrt factor recently milev author derived exact yield inscribed ellipse method the simplex number numerical improvements obtained compared general estimates known here compare yields real geometric method results complex pluripotential theoretical approach the simplex observe few remarkable facts comment existing conjectures formulate number hypotheses