Two Improved MQ Quasi-interpolation Operators

Based on the excellent properties of radial basis function,it has been successfully applied to neural network,digital image processing and numerical solution of partial differential equations.Radial basis function interpolation is an application of radial basis function.However,with the increase of the interpolation node number of radial basis function interpolation,it becomes very difficult to solve the coefficient matrix corresponding to the radial basis function interpolation.Sometimes,it may be an ill-conditioned matrix.In this case,the calculation of solving the coefficient matrix becomes unstable.Therefore,radial basis function quasi-interpolation is studied.The advantage of radial basis function quasi-interpolation is that it does not need to solve the system of linear equations.Moreover,some of quasi-interpolation operators have good properties such as polynomial reproducibility,monotonicity-preserving property and convexity-preserving property or concave-preserving property.The representative of radial basis function quasi-interpolation is Multiquadric(MQ)quasi-interpolation operators.In order to improve the approximation accuracy and property of the quasi-interpolation operator,in this paper,two improved MQ quasi-interpolation operators with good property are proposed.This paper has five chapters.The first chapter is the introduction.It mainly introduces the background of radial basis function and the research status of MQ quasi-interpolation,as well as it summarizes our main work.In the second chapter,some of preparatory knowledge is introduced.The related knowledge of radial basis function and radial basis function interpolation is summarized.Four MQ quasi-interpolation operators and their properties are also introduced.Meanwhile,we introduced three improved quasi-interpolation operators:based on the LD f(x)quasi-interpolation operator,Ling constructed quasi-interpolation operator LRf(x)by selecting two sequences point.Feng Renzhong constructed a good shape-preservation property and higher approximation quasi-interpolation operator Ldf(x).Wang Ziqiang constructed a new quasi-interpolation operator LRf(x)with cubic polynomial reproducibility and the strictly shape-preserving property to third and fourth derivative.In the third chapter,an improved MQ quasi-interpolation operator LdRf(x)is proposed.The new operator not only preserves the good quality of polynomial function but also inherits the approximation effect to the exponential function.Moreover,the operator LdRf(x)possesses quadratic polynomial reproducibility and strict cubic shape-preserving property.The results of numerical examples show that the new operator LdRf(x)has good approximation accuracy to power function,trigonometric function and exponential function.In the fourth chapter,an improved operator L*f(x)based on Ldf(x)is constructed.The numerical example shows that the new operator has good approximation.At the same time,the approximation effect of L*f(x)is better than LDf(x)and Ldf(x).What's more,the new operator has linear polynomial reproducibility.In the fifth chapter,it is the summary and prospect.We summarized the main content of this paper and the future work.