Fundamental Error Estimate Inequalities for the Tikhonov Regularization Using Reproducing Kernels

Abstract

First of all, we will be concentrated in some particular but very important inequalities. Namely, for a real-valued absolutely continuous function on [0,1], satisfying f(0)=0 and $\int_{0}^{1}f'(x)^{2}\,dx<1$, we have, by using the theory of reproducing kernels

$$\int_0^1\left(\frac{f(x)}{1-f(x)}\right)^{\prime\,2}(1-x)^2\,dx \le\frac{\int_0^1f^{\prime\,2}(x)\,dx}{1-\int_0^1f^{\prime\,2}(x)\,dx}. $$

A. Yamada gave a direct proof for this inequality with a generalization and, as an application, he unified the famous Opial inequality and its generalizations.

Meanwhile, we gave some explicit representations of the solutions of nonlinear simultaneous equations and of the explicit functions in the implicit function theory by using singular integrals. In addition, we derived estimate inequalities for the consequent regularizations of singular integrals.

Our main purpose in this paper is to introduce our method of constructing approximate and numerical solutions of bounded linear operator equations on reproducing kernel Hilbert spaces by using the Tikhonov regularization. In view of this, for the error estimates of the solutions, we will need the inequalities for the approximate solutions. As a typical example, we shall present our new numerical and real inversion formulas of the Laplace transform whose problems are famous as typical ill-posed and difficult ones. In fact, for this matter, a software realizing the corresponding formulas in computers is now included in a present request for international patent. Here, we will be able to see a great computer power of H. Fujiwara with infinite precision algorithms in connection with the error estimates.