Spline Quasi-interpolation And Multiquadric Quasi-interpolation Of Integral Value On Continuous Interval

The problem studied in this paper is how to use the information of integral value of unknown function to solve the function reconstruction.In practical application,we often deal with this phenomenon involving functions(28)(7)xfy(8)and their integral value functions.For example,in mechanics,acceleration and velocity;in statistics,probability density function and distribution function;in electricity,current function and charge function.In a word,this kind of problems often appear in mathematical statistics,mechanics,electricity,climatology,oceanography and so on,and have attracted wide attention[1-5].In recent years,many scholars are devoted to the study of this problem.However,in the early stage,the research method of integral spline interpolation needs to solve a large number of linear equations and additional boundary conditions,so the derivation process is relatively complex.The biggest advantage of spline quasi-interpolation is that it can directly give the approximation function without solving any linear equations.Moreover,spline quasi-interpolation has the advantages of good shape preserving,polynomial reproducing and stable calculation.Therefore,this paper presents a method to construct spline quasi-interpolation directly by using the information of integral value on continuous equidistant interval.Multiquadric quasi-interpolation is also a kind of very effective quasi-interpolation operators,which can also be used in approximation theory and numerical solutions of differential equations.Therefore,this paper also gives a method to construct multiquadric quasi-interpolation directly by using the information of integral value on continuous equidistant interval.This paper is divided into four parts: the first chapter introduces the problem,introduces the research status of integral value spline interpolation,integral value spline quasi-interpolation and multiquadric quasi-interpolation;the second chapter introduces the basic theory of B-spline,spline space and multiquadric quasi-interpolation operator;the third chapter proposes a new method of constructing spline quasi-interpolation based on the integral value given on continuous interval rather than the function value at node.Method,which is called integral type spline quasi-interpolation.This is a direct construction method without solving linear equations.In the fourth chapter,we propose a method to construct multiquadric quasi-interpolation directly by using the information of integral value on continuous equidistant interval.