On The Shape Preserving Properties Of Generalized Baskakov Operators

As a promotion of the Bernstein operator in the infinite domain, the researches of theshape-preserving properties of Baskakov operator also have important application value to otherdisciplines. In recent years, many scholars have conducted further researches on theshape-preserving properties of the Baskakov operators and have got similar conclusions with theBernstein operators, the shape-preserving properties of classical Baskakov operators are consistentto the Bernstein operators,see[1-3]. Zhang Chungou[3]mainly studies the Baskakov-Kantorovichoperator. Compared with classical Baskakov operators, the shape preserving properties of theKantorovich deformation operators have changed in some sense. In order to maintain the originalnature, we need to add some constraints.At the same time, he defined some new indicators tomeasure the shape preserving properties of the Baskakov operators, such as the star-properties,shape-preserving properties under average and so on. Thus to make a further description. Previousstudies on Bernstein operator mainly use the method of structure analysis, in view that theBaskakov operators are probability type operators, Liu Shenggui, Zhang Chungou and otherscholars[3-7]use probability method in the proof process. Liu Shenggui[4]uses probability theory toprove the shape preserving properties of Baskakov operator with parameters.This paper is based on the further study of the shape-preserving properties of classicalBernstein operators and Baskakov operators, uses the similar proof methods to conduct parallelextension to the new operators, researches the shape preserving properties of deformation operatorswhich contain parameters, as well as gives the definition and simple proof of two kinds ofdeformation of operators with two variables,thus to extend the application range of theshape-preserving properties.In the first chapter, we introduce the significance of research and theories which have beenformed already at home and abroad. In the second chapter, we use the samilar proof methods andconclusions in the process of proofing the shape-preserving properties of the Baskakov operators toproof the shape-preserving properties of a new Baskakov operator which contains parameter t inmonotonicity, convexity, smoothness, the Lipschitz function class. In the third chapter,we use newmethod to proof the shape-preserving properties of Baskakov-Durrmeyer operator. In the fourthchapter, we study a new class of Baskakov operator containing the parameter and give theapproximation properties of the operator; In the fifth chapter and the sixth chapter, we define twokinds of generalized Baskakov operators on a square domain and study their Lipschitz properties.