The Best Approximating Of Bounded Linear Operator With Spline And Its Application In Numerical Calculation

In this paper we have discussed the best approximating operator and the best approximating of bounded linear operator in H01[a,b] and H1[a,+∞). In Chapter Three, we use the method of interpolation spline of differential operater to come up with the reproducing kernel in H01[a,b] with respect to bounded linear operator in H10[a,b]. Then we use the reproducing kernel to develop the expression of the best approximating of bounded linear operator in H10[a,b] and prove its convergence. In Chapter Four, we generalize the method of interpolation spline of differential operater in the area of in H10[a,b] and H1[a,+∞) in a broader sense. The new methodof defining spline function will generalize the concept to the more universal Hilbert space. Hence, we take into consideration the interpolation spline function, general linear differential operater in semi- open interval and the Green function defined accordingly, and the reproducing kernel to produce the Green function and prove the existence of the reproducing kernel. As a result, we have demonstrated that the best approximating operator and the best approximating of bounded linear operator in H10[a,b] and H1[a,+∞) are consistent and, at the same time, we have generalized the theorem of the best approximating of bounded linear operator to the more universal H1[a,+∞) space.In Chapter Five, we compare numerical integration of the reproducing kernel with the integral equation of complexified trapezoid through a detailed solution to the initial-value problem of ordinary differential equation to exemplify that the method of interpolation spline integrating and the reproducing kernel is not only feasible in theory but also significant in pratice.