Modified Bernstein Operator And Spherical Radial Basis Function (rbf) Approximation

The operator approximation and spherical data approximation are the im-portant branches which have theoretical and practical significance in the modern function approximation theory. In this thesis,we mainly investigate approximation properties of modified Bernstein operators and spherical radial basis functions in-terpolation. The main contents of this paper are as follows:In chapter one,the background, history, significance and recent situation of Bernstein type operators and approximation on sphere are introduced, also some classical theoretical results related to this thesis are described.In chapter two, for the Bernstein type operators, using K-functional and high order modulus of smoothness, the direct and inverse theorems of approximation, and the equivalent theorem of high order approximation characterization are established. The results show that the effect of operator approximation characterization depends on the smoothness of the function.In chapter three, the approximation of minimum norm interpolation by the spherical radial basis function are investigated.Using relationship between the over-all and local of Spherical Duchon framework,we investigated the approximation of spherical functions with splitting spherical by spherical radial basis functions interpolation.The Lp error relationship between the minimum norm interpolation approximation and the target function is established.The results show that the order of approximation depends on the smoothness of the function.In chapter four,we summarize the main conclusions of the paper and put for-ward the questions and work in the further research.