The Construction Of Reproducing Kernel And Some Approximating Problems In The Reproducing Kernel Spaces

In this paper we have discussed a new method for constructing the reproducing kernels in the complicated reproducing kernel space W₂^m [a ,b], and the optimal approximating operator and the optimal approximating of the bounded linear operator. Moreover we have worked out the recursion formula of the reproducing kernel in W 2m [R ]. In the Chapter Three, we computed the reproducing kernel of the simple space W₂¹ [a ,b] by using the differential operator interpolation spline function. Furthermore, we verified that the method in computing the reproducing kernel by solving a boundary value problem for a differential equation directly actually is the special case of the method analyzed in this paper. Then we use the reproducing kernel to develop the expression of the optimal approximating of bounded linear operator in W 21 [a ,b] and proved the consistency between the optimal approximating operator and the optimal approximating of the bounded linear operator. In the Chapter Four, we generalized the new method into more common space W₂^m [a ,b], and worked out the appropriate Green function by L-function. As a result, we proved the existence of the reproducing kernel and developed the expression originally. At the same time, we proved that the new method is applicable in common reproducing kernel space. Then we obtained the same result of the consistency between the tow optimal approximating operators. In the end, we have demonstrated the recursion formula of the reproducing kernel in W 2m [R ] which made the computation easier and more convenient.In Chapter Five, we combine the reproducing kernel with the spline function through a detailed solution to the initial-value problem of differential equations with variable to exemplify that the method of interpolation spline integrating and the reproducing kernel is not only feasible in theory but also in practice.