The Approximate Properties Of Some Well-known Operators' Quasi-interpolants

The theory of operator's approximation mainly studies the convergent properties and convergence rate of a sequence of linear operators and other relative problems. The study of direct and inverse theorems and equivalent theorems of some well-known linear operators (such as Bernstein, Szasz-Mirakyan, Gamma, Baskakov operator and their modification of Durrmeyer and Kantorovich) is an important research program in the theory of operator's approximation, it is also significant in theory and application. The improvement of these operators' approximation order is one of subject. Thus the linear combinations of some operators were introduced and studied deeply, and a series of results have been achieved. Recently another way of improving approximation order has been suggested, in which some well-known operators' quasi-interpolants are introduced. Based on the earlier results, the present paper makes a study in the approximate properties of these well-known operators' quasi-interpolants, and obtain some results which are as follows:Firstly, by using of unified modulus (f,t), the direct and inverse theorems and equivalent theorems on Bernstein, Bernstein-Durrmeyer, Szasz-Mirakyan, Szasz-Mirakyan-Kantorovich quasi inter-polant operators for functions in the space C have been investigated, and the results include the result on Bernstein quasi-interpolant operator of P. Mache and A.T. Diallo and that on Szasz-Mirakyan quasi-interpolant operator of A.T. Diallo repectively.Secondly, the approximation equivalent theorem for Szasz-Mirakyan-Kantorovich quasi-interpolant operators have been studied by using of Ditzian-Totik modulus (f,t) in the space Lp, where 1