Around some extremal problems for multivariate polynomials (Q6632493)

Let \(E\) be a compact subset of \(C^N\) and \(V_E\) be the pluricomplex Green's function of \(E\). The authors consider the Hölder continuity property, which is one of the most interesting features of \(V_E\). By means of a radial modification of \(V_E\) they give some equivalent conditions to Hölder continuity property, connected with the Pleśniak property and the Markov inequality for polynomials.\N\NPolynomial inequalities have long been investigated in various norms. In the second section the authors present basic properties of extremal functions and capacities related to given norms. Moreover, they consider a capacity, a Chebyshev constant and a transfinite diameter with respect to a fixed norm on the space of polynomials of \(N\) variables and prove that this capacity is not greater than a corresponding Chebyshev constant.\N\NThe third section is devoted to another extremal problem of multivariate approximation. Namely, the authors consider an extremal family of polynomials that can be used in an economisation procedure and telescoping approximation series. This allows them to choose some elements from a sequence of approximation polynomials in order to get a faster approximation of functions.\N\NThe forth section deals with the relationship between Markov-type inequalities, Pleśniak property and the growth of the Green's function near a set \(E\). In particular, Corollary 4.4 shows that the Hölder continuity property and the Pleśniak property are equivalent.