Numerical harmonic analysis and its applications to image analysis and computational mathematics

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Outline :We have constructed new modified spline functions in multi-dimensions and have applied them to numerical analysis of nonlinear partial differential equations and image analysis. Also, we invited from abroad totally 10 active researchers in three years to organize international meetings. The details are summarized as follows.1.Computational harmonic analysis :We succeeded in construction of basis functions with "minimal" supports which enables rigorous interpolation and good sampling approximation of functions. Basis functions defined in more general domains are also investigated, i.e. in the case of finite intervals, several typical domains in the plane, e.t.c. To our surprise, almost radial basis functions have been obtained in several dimensions.2.Application lo numerical analysis of PDE :As an application, we have found a new method of computiong a series of finite difference schemes of higher orders to partial differential oparators. Also, we computed stable numerical solution to the important KP equation whose analytical solutions are difficult to find. The study of multi-scale phenomena and image analysis is ongoing.3.Harmonic analysis of complex singularity :In a joint work with oversea researchers, we developed the theory of harmonic analysis on a complex variety with singularities. This work will hopefully inspire us in future works related to singularities.