The Research About Baskakov Type Of Operators’ Approximation Property

Baskakov operators are the important tools which are used to research the problem ofoperators’ approximation. Because of its good approximation properties, Baskakov operators haveattracted a large number of experts to carry out a detailed study on it’s function approximationtheory. And all these researching results have provided Baskakov operators with wide application inthe field of function approximation theory.In this paper, some class of generalized Baskakov operators, q-Baskakov operators and theirdeformation operators’ approximation problems were studied. The generalized Baskakov operatorswhich were the comprehensive combination of Mirakyan-Szasz operators and the generalizedBaskakov operators were first proposed in the Fifth National Conference of approximation theory in1987by Chen Wenzhong. Since the q-Baskakov operators proposed by Aral and Gupta based onq-Bernstein operators, they have a lot of similar approximation properties. Research onthe deformation operator has a long history. The Baskakov type deformation operators haveapproached the study of the nature of many meaningful results. In foundation of existing researchresults. The strong converse inequality issues of a class of generalized Baskakov operators,approximation properties of a class of modified q-Baskakov operators, and the modifiedq-Baskakov-Szasz operators approximation properties were studied and discussed. The content ofthis paper is divided into the following several points:The first chapter, we firstly introduce the origin and the progress of the research on thefunction approximation theory, then we give an overview of research progress of the generalizedBaskakov operators, q-Baskakov operators and the deformation operators at home and abroad,finally some definitions and related notations are described used in this article are described, and themain content of this paper to study the overview.In chapter two, we investigate the strong inverse inequality problem of generalized Baskakovoperators used the knowledge of the K-functional and the moduli of smoothness and get a theoremconcerning strong inverse inequality.In chapter third, we defined the modified q-Baskakov operators and investigate the propertiesof approximation problems and gived the approximation order and some other conclusions of themodified operators used the knowledge of the modulus of smoothness and the K-functional.In chapter fourth, we discussed the approaching convergence of a class of q-Baskakovdeformation operators, studied and discussed the correction operator approximation order andconvergence rate of weighted approximation problems, through the introduction of a class of modified q-Baskakov-Szasz operator.In chapter fifth, we give a summary of the full text and an outlook into the generalizedBaskakov operator and q-Baskakov operator and the operator approximation problem.