Functional analytic and numerical study of partial differential equation and tomography

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(1)Liouville's theorem, famous in the theory of functions, was generalized by L.Schwartz in the form "there exists a solution of polynomial growth if and only if the total symbol of the equation has no real zeros", employing the tempered distributions. In this research I studied existence of solutions with quasi-exponential and infra-exponential growth, employing Fourier hyperfuncitons and its ultra-distribution analogues. Fourier hyperfunctions, contrary to Schwartz distributions, may have support only at infinity. Therefore the validity of Liouville type assertion depends on the approaching speed of the complex zeros to the real axis at infinity. (2)Generalizing the explicit construction by Morimoto-Yoshino of Fourier hyperfunctions supported by one point at infinity, I succeeded in constructing such objects in any quasi-exponential growth, employing the Ahlfors distortion theorem. (3)In collaboration with A.L.Balandin in Irkutsk, we gave a theory to reconstruct the solenoidal part of a 3D vector field from a small number of projection data of the inner product with the obserbed directions, by using expansion by vector spherical harmonics. (4)As a provisionary work to apply the theory of D-modules to error-correcting codes, we implemented a new example of algebraic-geometric codes, and applied it to color raster images. (5) As a provisionary work of application of microlocal analysis to image analysis, we developped socket communication protocol and GUI on unix Xlib, and using them made a graphical implementation of fair coin flipping, its Japanese version (janken), and mental poker protocol for three players. We also gave a definition of wave-front-set of images and analyzed its fundamental behaviour.