Studies Of The Equivalent Theorems Of Bézier-type Operators

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, 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. Several years before, Bezier-type operators were introduced and studied elementarily, but it is necessary to study these operators deeply with its using field wider and wider. Based on the earlier results, the present paper makes a study in the approximate properties of these Bezier-type operators , and obtain some results which are as follows:Firstly, by using of unified modulus ω_φλ(f,t)_∞ the pointwise direct and inverse theorems and equivalent theorems on Baskakov-Bezier, Bernstein-Bezier and Szasz-Durrmeyer-Be'zier operators have been investigated.Secondly, the approximation equivalent theorem for Bernstein-Kantorovich-Be'zier operators have been studied by using of smooth modulus ω_φ(f,t)_p in the space L_p(1 ≤ p≤∞).Thirdly, for Bernstein-Durrmeyer-Bezier operators, we have given the approximation equivalent theorem in L_p(1 ≤ p≤∞).