基于p进Vilenkin群小波方法解微分方程

The wavelet methods for differential equations on the p-adic Vilenkin groups

定义了基于p进Vilenkin群的傅里叶-沃尔什变换,介绍了Haar小波函数系,多分辨分析和函数的(拟)Haar表示方法以及函数的沃尔什表示形式.最后研究了如何利用函数的沃尔什表示和Gibbs导数的性质证明n阶线性微分方程和一阶齐次波动方程解的存在唯一性.

In this paper, Fourier-Walsh transform is defined based on p-adic Vilenkin groups . Furthermore, Haar wavelets, multiresolution analysis and （quasi-Haar） Haar representation of functions as well as the representation of Walsh functions are introduced. Finally, an algorithm for the existence and uniqueness of solutions of a linear differential equations of order n and the one-dimensional homogeneous wave equation are presented.