POSITIVE BERNSTEIN-SHEFFER OPERATORS

Let h(t) = Sigma(n greater than or equal to 1) h(n) t(n), h(1) > 0, an d exp(xh(t)) = Sigma(n greater than or equal to 0) P-n(x) t(n)/n!. For f is an element of C[0,1], the associated Bernstein-Sheffer operator of degree n is defined by Bi:f(x)= P-n(-1) Sigma(k = 0)(n) f(k/n)((n)( k)) P-k(x) P-n - k(1 - x) where p(n) = p(n)(1). We characterize functi ons h for which B-n(h) is a positive operator for all n greater than o r equal to 0. Then we give a necessary and sufficient condition insuri ng the uniform convergence of B-n(h) f to f. When h is a polynomial, w e give an upper bound for the error parallel to f - B(n)(h)f parallel to(infinity.) We also discuss the behavior of B-n(h) f when h is a ser ies with a finite or infinite radius of convergence. (C) 1995 Academic Press, Inc.