Functional analytic and numerical study of partial differential equation and tomography

(1) L.Schwartz将函数论中著名的Liouville定理推广为“当且仅当方程的总符号没有实数零存在多项式增长的解”，采用了缓律分布。本文利用傅立叶超函数及其超分布类似物研究了拟指数和次指数增长解的存在性。傅里叶超函数，与Schwartz分布相反，可能只有在无穷远处才有支持。因此，刘维尔型断言的有效性取决于复零在无穷远处接近实轴的速度。(2)推广了Morimoto-Yoshino关于无穷远处有一点支持的傅里叶超函数的显式构造，利用Ahlfors畸变定理，成功地在任何拟指数增长中构造了这类对象。(3)与伊尔库茨克的all balandin合作，提出了一种利用矢量球面谐波展开，利用少量内积与观测方向的投影数据重构三维矢量场螺线线部分的理论。(4)作为将d模理论应用于纠错码的初步工作，我们实现了一个新的代数-几何码示例，并将其应用于彩色光栅图像。(5)作为将微局部分析应用于图像分析的临时性工作，我们在unix Xlib上开发了套接字通信协议和GUI，并利用它们实现了公平抛硬币及其日文版本（janken）和三人心智扑克协议的图形化实现。给出了波前图像集的定义，分析了波前图像集的基本性质。

(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.