Global Smoothness Preservation by Multivariate Operators

In this chapter we discuss the global smoothness preservation by some multivariate approximating operators. By extending a fundamental result of Khan and Peters, we establish a general result for operators having the splitting property. Furthermore, we show more complete inequalities for Bernstein operators on the k-dimensional simplex and cube, formulate a certain transfer principle for tensor product operators, and apply an earlier related result in the context of stochastic approximation. Here we follow the basic study done by the first author, Cottin and Gonska [23].