Reverse Martingales and Approximation Operators Original Research Article

Let {ξn, Fn, n ≥ m ≥ 1} be a reverse martingale such that the distribution of ξn depends on x ∈ I ⊂ R =(− ∞, ∞)x. for each n ≥ m, and ξn[formula] For a continuous bounded function f on R let Ln(f, x) = Ef(ξn) be the associated positive linear operator. The properties of ξn are used to obtain the convergence properties of Ln(f, x), and some more details are given when ξn is a reverse martingale sequence of U-statistics. Lipschitz properties for a subclass of these operators resulting from an exponential Family of distributions are also given. It is further shown that this class of operators of convex functions preserves convexity also. An example of a reverse supermartingale related to the Bleimann-Butzer-Hahn operator is also discussed.