Q-for Sz¨¢sz The-mirakjan Operator

By means of the q-integers as well as of the Gaussian binomial coeffcients, G.M.Phillips[16] and T.Trif[18] introduced respectively the q-Bernstein polynomials B_n,q and the q-Meyer-Konig and Zeller operators M_n,q both generalizing classical ones. It was found that the q-Bernstein polynomials and the q-Meyer-Konig and Zeller operators possess many remarkable properties (see[5-6][9-14][16-18][20-21]).In this dissertation ,we introduce the sequence of q-operators {S_n,q}_n≥1(q > 0,q ≠ 1), depending on q-integers and q-analogue of e^x,which generalize the sequence of classical Szasz-Mirakjan operators {S_n,q}n≥1 Basic properties of the new q-operators are given. Also,shape-preserving properties, convergence properties, the rate of convergence of {S_n,q}n≥1(0 < q < 1) to S_∞,q are obtained.