Research On Approximation Properties Of Q And (p,q) Probability Operators

The q type and (p,q) type of the operators are used in the file of the probability operators due to the good property,especially the Bernstein operators,the Kantorovich operators and the Gamma operators.Therefore the experts take more and more time to learn the operators,which make the file of the operators more and more sufficient to be the important tool of study approximation problem.This paper mainly discusses the approximation property of some q type and(p,q)type of the operators,which focus on the Szász-Kantorovich operators,the Bernstein-Kantorovich operators and the Gamma operators.It mostly includes the modified q-Szász-Kantorovich operators,the modified(p,q)-Szász-Mirakjan-Kantorovich operators,the stancu of (p,q) Gamma operators,the generalized(p,q)-Gamma-Stancu operators and the modified(p,q)type of the Bernstein-Kantorovich operators,and then we conclude into six parts.Firstly,we review the research background of operators approximation theories which contains the details of the Mirakjan type and the Kantorovich type of the q-Szász operators,the q-Gamma operators and(p,q)-Gamma operators at home and abroad.And Finally we also introduce some symbols and definitions of the paper as well as main contents.Secondly,we maily construct the modified q-Szász-Kantorovich operators based on the q-Szász-Mirakjan operators and the q-Bernstein-Kantorovich operators.Also we obtain the approximation propertities by the smooth modulus.Thirdly,The modified(p,q)-Szász-Mirakjan-Kantorovich operators are gained on the basis of the(p,q)-Szász-Mirakjan operators of T.Acar and the second chapter.Then we conclude the approximation problem of the weighted space.Fourthly,We add the new parameter to make the modified(p,q)-Bernstein-Kantorovich operators based on the q-Bernstein-Kantorovich operators.At the same time,we also use the Steklov mean and the smooth modulus to study the approximation properties.Fifthly,We mainly construct a new generalized(p,q)-Gamma-Stancu operators to gain the point estimate by using the continuous modulus and other tools,Finally we also discuss the local approximation properties and the other approximation problems of the operators.Sixthly,We define a new kind of (p,q)-Gamma-Stancu operators to infer the Voronovskaja type theorem and the rate of the convergence,giving the asymptotic formulas of the origin moments and the central moments.Seventhly,We summarize the all paper and discuss the further prospects of the modified q-Szász-Kantorovich operators,the modified(p,q)-Szász-Mirakjan-Kantorovich operators,the(p,q)-Gamma-Stancu operator,the Generalized(p,q)-Gamma-Stancu operators and the modified(p,q)-Bernstein-Kantorovich operators.