Interpolation Methods In The Two Given Reproducing Kernel Hilbert Spaces

The study of numerical approximation, known as a basic subject of computational mathematics, is a classic mathematical problem. It is noted that the study and application of numerical method is meaningful and useful in the domains of engineering technique. In this thesis, the problems of numerical approximation are discussed by virtue of the special theory of the reproducing kernel functions in the two given reproducing kernel Hilbert spaces.Firstly, a new method to resolve numerical approximation is introduced in a reproducing kernel Hilbert space H K[ a , b ] by use of the ideas of reproducing kernel space. Namely, interpolation function is constructed by base functions which are presented by the linear combination of a reproducing kernel function. In this thesis, best approximation, convergence and error estimate of interpolation function are discussed. Meanwhile numerical examples show the effect of interpolation method. The method is much easy for calculation: One hand, order of error estimate is convenient to discuss. On the other hand, during the process of interpolation, the coefficients of interpolation function can be obtained by reproducing kernel without solving system of equation.Secondly, another kind of means of numerical approximation is given in the space of the image space of wavelet transform based on the relationship between wavelet transform and the theory of reproducing kernel: Modulation Gaussian Function is introduced as a wavelet generating function, which is frequently used by R. Kronland-Martient and J. Morlet. The expression of the reproducing kernel function of the image space of this wavelet transform, and the isometric identity of the wavelet transform are given concretely. The structure of the image space of the wavelet transform is analyzed clearly. Moreover, interpolation formula is given so that wavelet transform at any point can be reconstructed by reproducing kernel function. These provide the theoretic basis for discussing the actual application of wavelet transform in the engineering.By way of conclusion, numerical approximation using the special theory of the reproducing kernel functions in the two given reproducing kernel Hilbert spaces is discussed thoughly in this paper. This provides fundamental theoretic basis for the numerical approximation of general reproducing kernel Hilbert spaces.