Approximation Of Lipschitz Continuous Functions By ?-Bernstein Operators

As the methods and ideas of functional analysis are integrated into the approximation theory,various types of operators with different shapes and different properties provide powerful expressions for the approximation tools.The operator approximation theory has developed rapidly and has become one of the hotspots of function approximation research both at home and abroad.In this thesis,we focus on the study of ?-Bernstein approximation of Lipschitz function class.The full text consists of four parts and the specific contents are as follows:Firstly,the origin of the function approximation problem and the development status at home and abroad are summarized.Classical method in operator approximation theory is emphasized: Bernstein operator approximation.It mainly includes the definition of the Bernstein operator,the approximation property,and its relationship with the Lipschitz function class.The second chapter mainly introduces a new class of generalized Bernstein type operators,namely ?-Bernstein operators and their approximation properties.This operator introduces an adjustable shape parameter based on the Bernstein operator,which not only includes the classical Bernstein operator,but also has excellent characteristics consistent with the Bernstein operator.At the same time,the introduction of the tunable parameter ? makes the function approximation method more flexible and effective.The third chapter is the main work of this thesis.The relationship between ?-Bernstein operator and Lipschitz function class is studied and proved.The ?-Bernstein polynomial belongs to the Lipschitz function class and has the same Lipschitz constant.That is to say,if the approximation function belongs to the Lipschitz function class,the corresponding ?-Bernstein operator also belongs to the Lipschitz function class;conversely,if the ?-Bernstein polynomial belongs to the Lipschitz function class,the approximation function also belongs to the Lipschitz function class.Finally,the thesis is summarized.According to the development hotspots of current operator theory,the research direction and content of the next step are proposed,and the feasibility is analyzed.