Proposed Several Interpolation Theory And Its Application

Quasi-interpolation plays a vital role in approximation theory and its applications. One major advantage of quasi-interpolation is that it can yield an approximant directly without the need to solve any large-scale linear system of equations. Moreover, some quasi-interpolation can even possess shape-preserving properties (such as the MQ quasi-interpolation, the spline quasi-interpolation, etc). Besides, compared with interpolation, quasi-interpolation has some other advantages such as stability of computation, a small amount of computation, etc. Partic-ularly, when the sampling data are noised, quasi-interpolation can also filter the noise. Thus, quasi-interpolation has been studied extensively both in theory and in practical applications.However, most studies of quasi-interpolation are usually for the case when the sampling data are discrete function values (or a finite linear combination of discrete function values). Note that in practical applications, more commonly, the sampling data are linear functional data (the discrete values of the right-hand side of some differential equation) rather than the discrete function values (such as solving differential equations, remote sensing, seismic data, etc). Therefore, to make quasi-interpolation more available for practical applications, it is more meaningful to discuss quasi-interpolation for the linear functional data.On the other hand, when solving the numerical solution of a differential equation with quasi-interpolation, one usually use quasi-interpolation to approximate high-order derivatives. This will reduce the approximation order of the numerical solution. Thus, to get a numeri-cal solution with high approximation order, one needs to construct such a quasi-interpolation scheme that can provide the numerical solution directly from the discrete values of the right-hand side of the equation coupled with the boundary conditions (initial conditions or others) without the need to approximate high-order derivatives.Based on the above two points, this dissertation mainly studies the construction of quasi-interpolation for the linear functional data and its applications.We construct a quasi-interpolation scheme for the linear functional data and give its error estimate. Based on the error estimate, one can find a quasi-interpolant that provides an optimal approximation order with respect to the smoothness of the right-hand side of the differential equation. As applications, we apply it in solving the numerical solutions of differential equations and constructing the Lyapunov function in dynamical systems, respectively.Both theory and numerical results show that the scheme can overcome the drawbacks of the meshless collocation method:solving large-scale linear system of equations, unstability of computation, etc.In some cases, the sampling data usually possess periodicity, such as signal processing, medical image processing, etc. Then, obviously, it is more reasonable to fit the data with a periodic function.Trigonometric B-splines quasi-interpolation can fit the periodic sampling data. However, the order of smoothness of trigonometric B-splines quasi-interpolation is low, since it uses trigonometric B-splines as its kernel functions. Thus, if approximating high-order derivatives with trigonometric B-splines quasi-interpolation, one should use high-order trigonometric B-splines as the kernel functions. This implies computation of high-order generalized divided differences, which leads to the unstability of computation.To overcome the drawbacks of trigonometric B-splines quasi-interpolation, a periodic quasi-interpolation scheme with infinite smoothness is constructed in the last part of this dis-sertation. The scheme can not only fit the periodic sampling data, but also approximate high-order derivatives. Moreover, since the construction of the kernel functions of the scheme only requires second-order generalized divided differences, it avoids the unstability of high-order generalized divided differences.As applications, we apply it in approximating a function, the first-order derivative, the second-order derivative of the function, and solving the numerical solution of time-dependent partial differential equations, respectively. Numerical results show that the scheme can not only provide an excellent approximation to the function, but also give an approximation to derivatives of the function.