Global smoothness preservation by multivariate Bernstein-type operators (Q2702485)

The authors consider a very general class of multivariate Bernstein-type operators representable by means of suitable stochastic processes. (Twelve families of one-dimensional operators have been collected in the Appendix and explicitely represented by using a few types of well-known stochastic processes). For such operators two problems are considered: (i) preservation of the usual \(l_p\)-modulus of continuity, and (ii) preservation of classes of functions determined by moduli of continuity. Problems of this type have been discussed by several authors in the recent past, as can be read about in the historical notes at the end of the paper. Using a probabilistic approach based on representations in terms of stochastic processes, the authors develop a general theory concerning the constants involved in the problems (i) and (ii). General bounds and formulae for the best constants are given, with special attention paid to tensor product operators and to simplicial operators. Applications and (graphical) illustrations of the general results are also given, showing that, when dealing with concrete operators, one obtains the exact values of the constants or at least further insights into the problem under consideration.NEWLINENEWLINEFor the entire collection see [Zbl 0954.65001].