Correction Of A Rational Operator Approximation Theorem

The approximation theory is one of the important branches of modern mathematics, longin history, rich and practical in contents. It began in19th century when two famous theoremswere established: Weierstrass established the theorem on approximation by polynomials ofcontinuous functions in1885and Chebyshev established the theorem on characterizationfor approximation by polynomials in1859. It grew prosperous in the20th century andbecame an independent subject. As a special tool of nonlinear approximation, the family ofrational operators is a special class. Because of its construction and roles, it attracts manymathematicians’ attentions. In this thesis, we divide our study into two parts. Firstly, wedefine the Durrmeyer modified rational operators, and divide it between the integral modifiedrational operators with geometric domination and the integral modified rational operatorswith arithmetic domination by corresponding domination function of the kernel function.Furthermore, we give their approximation theorem. Secondly, we study the approximationproperties of Shepard operators based on the Jacobi weight function in Cwspace.This thesis can be made up of four chapters.In the first chapter, we give some relative conceptions, notations and fundamental the-orems and introduce the definition and some results of the Bak operators, the Shepard oper-ators, the Vertesi operators, the Nevai operators and their Kantorovich modified operatorsand Durrmeyer modified operators.In the second chapter, we discuss the characteristics, approximation theorems and ap-plications of the integral modified rational operators with geometric domination and theintegral modified rational operators with arithmetic domination.In the third chapter, we study the approximation properties of Shepard operators basedon the Jacobi weight function in Cwspace.In the fourth chapter, we give the conclusions of the thesis.