On Approximation by Post-Widder and Stancu Operators Preserving x2

In the papers [5]-[7] was examined approximation of functions by the modi edSz asz-Mrakyan operators and other positive linear operators preserving e 2 (x) = x 2 : In thispaper we introduce the Post-Widder and Stancu operators preserving x 2 in polynomialweighted spaces. We show that these operators have better approximation properties thanclassical Post-Widder and Stancu operators. 1. Introduction1.1. The Post-Widder operators(1) P n (f;x) P n (f(t);x) :=Z 10 f(t)p n (x;t)dt; x 2I; n 2N;(2) p n (x;t) :=(n=x) n t n 1 (n 1)!exp ntx ;I = (0;1), N = f1;2;g , were examined in many papers and monographs (e.g.[4]) for real-valued functions f bounded on I. It is known ([4], Chapter 9) that P n are well de ned also for functions e k (x) = x k , k 2N 0 = N [f0g, and(3) P n (e 0 ;x) = 1; P n (e 1 ;x) = x; P n (e 2 ;x) =n+ 1nx 2 and generally(4) P n (e k ;x) =n(n+ 1) (n+ k k1)xn k ; k 2N; Corresponding author.Received 19 June 2007; accepted 23 October 2007.2000 Mathematics Subject Classi cation: 41A25, 41A36.Key words and phrases: Post-Widder operator, Stancu operator, polynomial weightedspace, approximation theorem.