The Approximation For Linear Combinations Of Szász-Type Operators

In this thesis,we study strong type of Steckin inequality and the lower estimate for linear combinations of Szász-Mirakjan operators on space[0,?).The saturation class,strong type of Steckin inequality and the lower estimate for linear combinations of Szász-Kantorovich operators on space?0,???1?<??.In chapter 1,we introduce the backgroud about this thesis,then propose the main problems and give some correlatively basic concepts and symbols.In chapter 2,we investigate the inverse approximation problem for linear combina-tions of Szász-Mirakjan operators in saturation state,and establish its strong type of Steckin inequality,which includes the saturation class for linear combinations of Szász-Mirakjan operators.The lower estimate for linear combinations of Szász-Mirakjan op-erators including its direct and inverse theorems and saturation class are established by using the equivalent relations between K-functionals and modulus of smoothness for Ditzian-Totik and its strong type of Steckin inequality.In chapter 3,we construct a new K-functionals by using a differential operator gen-erated by Szász-Kantorovich operators,and give the exact expressions for the moments of each order of Szász-Kantorovich operators.On this basis,the saturation class for lin-ear combinations of Szász-Kantorovich operators on space?0,???1?<??is given and proved.In chapter 4,on the basis of the third chapter,we establish the strong type of Steckin inequality for linear combinations of Szász-Kantorovich operators,and then we study the relationship between the approximation speed for linear combinations of Szász-Kantorovich operators and the smoothness of approximated functions,giving the lower estimate for linear combinations of Szász-Kantorovich operators,which covers its direct and inverse theorems and saturation class.