The Simultaneous Approximation Properties By Baskakov-type Operators In H(o|¨)lder And Complex Spaces

The theory of operator approximation is the study of the convergence property and the quantization of the rate of convergence of linear operator sequences.It was established by P.P.Korovkin and H.Bohman for the continuous function in 1951,followed by many scholars extended to measurable function space,bounded variation function space,-integrable function space and other real function spaces and complex space.The theory of operator approximation is an important branch of the theory of function.It has a great significance in function approximation theory and has been widely application in many practical problems,such as signal imitation,image processing,automobile and aircraft manufacturing,etc.In this paper,we mainly study the approximation properties of Baskakov type operators in H(o|¨)lder and complex spaces.We give the definition of the discussion and improve the definition form of operators.Using the equivalence relation between smoothing modulus and K-functional,by recursion formula,the mean value theorem,the Bernstein inequality,the Taylor expansion and the high order Cauchy integral formula,we obtain the approximation order of Baskakov type operators in two spaces by analysis and synthesis.This paper is divided into four chapters.In chapter 1,the definitions of Baskakov type operators and several basic concepts such as space and norms which we needed are given.In chapter 2,firstly,we briefly introduce the properties of H(o|¨)lder space and the approximation theorem,then obtain the simultaneous approximation of the operators in H(o|¨)lder space through the recursion formula of Baskakov type operators.In chapter 3,we first introduce the properties of B(?)zier type operators,using the mean value theorem,we study the approximation property of the operators in H(o|¨)lder space and obtain the approximation result.In chapter 4,we briefly introduce the result of approximation and the new definition of Baskakov-Kantorovich operators in complex space,using the high order Cauchy integral formula,the asymptotic estimation and equivalence theorem of the operators in the complex space are obtained.