JOURNAL OF APPROXIMATION THEORY 1,215278 (1968) On Approximation Operators of the Bernstein Type D. LEVIATAN Department of Mathematics, University of Illinois, Urbana, Illinois 61801 1. INTRODUCTION Let the sequence {&}(i > 0) satisfy O<X~<X,<...<h,<...co, * 1 c I=1 x=m’ U-1) and define (the divided difference) [xAm,. . .) XAm] = izm Xh’/w~fn(Xi), OGmGn=O, 1,2, . . .. wherew,,(x)=(x--h,)...(x-h,),O,(m~n=0,1,2,....Denote p&x) = (-l)“- l A”, . . . A,- 1 [xAm,. * *, xAn], 0 < m -c n,p&x) = xA” and ix,, = ((1 - X,/X,). . . (1 - Xl~h”-,)}“~‘, O<m<n,cz,,= 1. Itiswellknownthatp,,(x)~OforO~m~n=0,1,2,...andO~x~1. With a functionf(t), bounded in [0, I], we associate the operators These operators, which generalize the Bernstein power-series of Meyer-K&rig and Zeller [7], were first introduced by Jakimosvki and the author in [4], where some approximation properties were stated without proof. (For proofs see [5]). More recently, these operators were redefined and their approximation properties, were studied independently and from a different point of view by Feller [2]. It is our purpose here to discuss the approximation properties of the derivatives of L,(f, x). 2. AUXILIARY LEMMAS We make use of the following lemmas. 19 275