Integro Spline Interpolation And Quasi-interpolation

Traditionally,we often use the function values at nodes and their derivatives to reconstruct function when we need to solve interpolation problem.However,with the development of science and technology,the interpolation problem is supposed to the solve more complex problems.For example,we deal with situations or phenomena which only involve the integral values of a given univariate function over successive subintervals rather than the function values at the knots.This problem has wide applications in the fields of numerical analysis,mathematical statistics,environmental science,mechanics,electricity,climatology,oceanography and so on.This paper is divided into five parts:the first chapter introduces the research status of integral spline interpolation and Quasi-interpolation at home and abroad;the second chapter mainly introduces some basic knowledge about spline space and B-spline space;In Chapter 3,we prove the existence and uniqueness of even-degree spline which interpolates given integral values,and we analyze smoothness of such even degree spline.In Chapter 4,we give a simple method to reconstruct function with the integral values of successive subintervals.Firstly,the function values at the knots can be given by the linear combination of the integral value.Then the approximated function values are plugged into the values of linear functional in discrete spline quasi-interpolation operators and so-called integro spline quasi-interpolation is constructed.