On Bernstein-Schnabl operators on the unit interval

Bernstein-Schnabl operators were first introduced by R. Schnabl in 1968 in the context of sets of probability Radon measures on compact Hausdorff spaces. Subsequently Grossman proposed a general method of constructing Bernstein-Schnabl operators on an arbitrary convex compact subset of a locally convex space and he showed that they are an approximation process for continuous functions. A particular class of these operators has been also studied by the F. Altomare and, subsequently, by several other authors. Their construction essentially involves positive projections and they satisfy many additional properties useful for the study of evolution problems. In this paper we deep the study of the Bernstein-Schnabl operators associated with a general continuous selection of probability Borel measures on the interval [0,1], which not necessarily arise from a positive projection. These operators seem to have some interest because they furnish new general approximation processes for continuous functions and they also approximate the solutions of the initial-boundary problems associated with a class of degenerate diffusion equations. In the first section we recall their definition and discuss some examples of them. After that, we investigate their approximation properties and show several estimates of the rate of convergence by means of suitable moduli of smoothness. Shape preserving properties are discussed in Section 2. In particular, we investigate some conditions under which these operators preserve the convexity. In the third section we show that suitable iterates of Bernstein-Schnabl operators converge to a Markov semigroup on C([0,1]) whose generator is a degenerate differential operator of the form Au(x):=\alpha(x) u''(x) (x \in [0,1] defined on a suitable subspace of smoot functions satisfying the so-called Wentcel boundary conditions. By means of Bernstein-Schnabl operators we establish some qualitative properties of this semigroup and, in particular, its asymptotic behaviour. In the same section we also study the generation properties of general differential operators and determine suitable continuous selections of Borel measures such that the iterates of the corresponding Bernstein-Schnabl operators converge to the given Markov semigroup.