Research of the applications of the theory of reproducing kernels

我们建立了再生核理论在Tikhonov正则化中应用的一般理论，并作为一般理论的应用，在反问题中得到了许多具体结果。典型的结果是热传导的有效反演公式和拉普拉斯变换的真实的反演公式，它们是非常著名的不适定问题，并且是众所周知的非常困难的问题。我们成功地在他们的数值分析，并显示在计算机图形与我们的相关合作者，特别是教授Tsutomu Matuura和Hiroshi Fujiwara。此外，我们能够在主要的国际期刊上发表他们的结果。我们在常微分方程、波动方程、Poisson方程、Dirichlet问题等方面得到了具体的结果。

We established a general theory of the applications of the theory of reproducing kernels to the Tikhonov regularization and derived many concrete results in inverse problems as applications of the general theory. typical results are the effective inversion formulas for the heat conduction and the real inversion formula of the Laplace transform that are very famous ill-posed problems and are well-known as very difficult problems. We succeeded in their numerical analysis and showed the results in the computer graphics with our related collaborators-in particular, Professors Tsutomu Matuura and Hiroshi Fujiwara. Furthermore, we were able to publish their results in which are major international journals. We derived the concrete results in ordinary differential equations, the wave equation, Poisson equation, Dirichlet problem, and so on.