The Approximation Of Q-Bernstein Type Operators And The Error Of Numerical Intergration

Approximation theory consists of the function approximation and the numerical approximation. The function approximation mainly uses simple functions to approach the general functions, while numerical approximation uses simple calculation to approximat- ively calculate complex calculations.On the function approximation, because Bernstein operator has good approximated nature and conformal nature, the relevant operators are favored by many scholars, and there are many promoted forms. Over these years, the research introducing q-integers for obtaining a variety of q-operators has attracted many scholars, people have studied the relative natures of a variety of q-Bernstein type operators, the studies show that the nature of q-Bernstein operators is different from the one of the classical Bernstein operators. There are many connections and differences between Bernstein operators and q-Bernstein operators, they have separated advantages. It is very significant to further study q-Bernstein type operators and compare the similarities and differences between the two approximation properties.About the numerical approximation, as we all known, the integral operation is a complex operation. On the one hand, in many cases the original function of the integrand is difficult to obtain, and even some functions does not exist the limited form original functions; The other hand, in many practical problems, it often only get some function values of the integrand at a number of discrete points, the result is that we can not find the original function to calculate the definite integral. Therefore, numerical calculation of the definite integral is very significant on the aspects of theory and practice. The quadrature formula commonly uses the trapezoidal formula and Simpson formula.The main work of the paper:Firstly, the paper discusses q-operator approximation of the real semi-axle contin- uous function. Based on Stancu-Chlodowsky operators, it introduces q-integers, and obtains q-Stancu-Chlodowsky operator, discusses the approximation nature and the conv- ergence rate of the operator, at last, the paper gives its Voronovskaya type theorem.Secondly, the paper discusses error estimation of numerical integration. On the one hand, it discusses the convergence rate that the complex Simpson formula with a certain smooth function approaches Riemann integral, and then using r-order Wiener measure to discuss the average error of the trapezoidal quadrature formula on the mearsure, and then draw the average error among all forms of nodes trapezoidal quadrature formula in the sense of the measure the equidistant nodes is optimal.