Generalized Szász Operators for the Approximation in the Complex Domain

Abstract

converge (as u -oo) to f (x) at each point t = x > 0, where f (t) is continuous. J. J. Gergen, F. G. Dressel and W. H. Purcell [2] proved that for a certain class of analytic functions f (z) the operators P(u, f ) approximate these functions. E. W. Cheney and A. Sharma [1] showed that the operators P(n, f ), n = 1, 2, , are variation-diminishing in the sense of I. J. Schoenberg [8]. They also proved that if f is convex, then P(n, f) is decreasing in n, unless f is linear (in which case P(n,f) = P(n + 1,f) for all n). Recently A. Jakimovski and D. Leviatan [4] considered the following generalization of P(u, f ): Let g(z) Z= o aj? be an analytic function in the disk lzl 1, and suppose g(1) = 0. Define the Appell polynomials Pk(X) pk(X, g), k > 0, by g(u)eux Pk(X)Uk. k = O