Reverse martingales and approximation operators
Let \((\Omega, {\mathcal F}, P)\) be a probability space and let \(\{{\mathcal F}_ n, n\geq 1\}\) be a decreasing sequence of \(\sigma\)-fields of sets in \(\mathcal F\). A sequence of random variables \(\{\xi_ n, n\geq 1\}\) such that \(\xi_ n\) is \({\mathcal F}_ n\)-measurable and \(E|\xi_ n|< \infty\), is called a reverse martingale, if for each \(n\geq 1\), \(E(\xi_ n| {\mathcal F}_{n+ 1})= \xi_{n+ 1}\) a.s. Under the assumption that the distribution of \(\xi_ n\) depends on \(x\in I\) and that \(\{\xi_ n, n\geq 1\}\) converges to \(x\), the author introduces positive linear operators \(L_ n\), defined on the space \(C_ B(R)\) of continuous bounded functions on \(R\) associated with the reverse martingales, by \(L_ n(f, x):= Ef(\xi_ n)\). Then applying ideas and results from the theory of reverse martingales he is able to obtain various approximation properties of these operators which include some well-known approximation operators like those of Bernstein, Szász, Weierstrass and Bashakov. The author finally applies his results to the Bleimann-Butzer-Hahn operators.
scientific article; zbMATH DE number 756633