CONSTRUCTIVE THEORY OF FUNCTIONS, Varna 2002 (B. Bojanov, Ed.), DARBA, Sofia, 2003, pp. 184-189. The Complete Asymptotic Expansion for the Gamma Operators and Their Left Quasi-Interpolants Ulrich Abel and Mircea Ivan Our purpose is to study the local rate of convergence of the Gamma operators. The talk presents the complete asymptotic expansion for the Gamma operators. We investigate their asymptotic behavior also con- cerning simultaneous approximation. All expansion coefficients are ex- plicitly calculated. It turns out that Stirling numbers play an important role. Moreover, we deal with linear combinations of Gamma operators having a better degree of approximation than the operators themselves. Using divided differences we define general classes of linear combinations, special cases of which were recently introduced and investigated by other authors. Finally, we study the left quasi-interpolants of the Gamma op- erators in the sense of Sablonni` ere. 1. Introduction M¨ uller’s Gamma operators are given by G n f (x) := x n+1 n!  ∞ 0 t n e −xt f  n t  dt (n =1, 2,... ) (1) for all functions f : (0, ∞) → R for which the integral on the right-hand side of (1) exists for all x ∈ (0, ∞). These operators have been introduced in [7] and investigated in subsequent papers [6], [8] and [10]. Newer results on Gamma operators can be found in [5] and [3]. The purpose of this paper is the study of the local rate of convergence of the operators (1). We investigate their asymptotic behavior also for simultaneous approximation. As main result we derive the complete asymptotic expansion (G n f ) (r) (x) ∼ f (r) (x)+ ∞  k=1 c (r) k (f ; x) n k (n →∞) , provided f possesses derivatives of sufficiently high order at x and satisfies certain growth conditions. Furthermore, we study general classes of linear