Research of the applications of the theory of reproducing kernels

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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.