Power Series of Bernstein Operators and Approximation of Resolvents

Abstract

According to the theory developed by F. Altomare and his school, certain C 0-semigroups can be approximated by iterates of positive linear operators. A. Albanese, M. Campiti and E. Mangino [J. Appl. Funct. Anal. 1 (2006), 343-358] proved that the resolvent (λ−A)−1 of the infinitesimal generator of such a semigroup can be also approximated, for λ > 0, by suitable iterates. What happens when $${\lambda \to 0^{+}?}$$ We give an answer in the case of the semigroup approximated by the classical Bernstein operators B n on the canonical simplex S of $${\mathbb{R}^{d}}$$ . Specifically, we show that $$-A^{-1}h = \lim\limits_{n \to \infty}\frac{1}{n}{\sum\limits^{\infty}_{k=0}}{B^{k}_{n}h}$$ for h in a certain subspace of C(S). This gives a new method to investigate the qualitative properties of the inverse of A.