Walsh Functions Of Quasi-Walsh Order And Its Applications

This dissertation has mainly studied Walsh functions of a new kind of order, which is called quasi-Walsh order. Some important properties of this kind of Walsh functions are examined and a few fast algorithms of Walsh transforms based on this order are designed. One special application is also discussed. According to the needs of this research, it has been systematically introduced the basic theory and the latest applications of Walsh functions. The main works and results of this dissertation can be summarized as follows:1. The basic theory of Walsh functions is systematically re-examined. This involves the general mathematic definitions of Walsh functions of five traditional orders, the evolving generations of them based on the copying and the bisection evolution technology respectively. The recursion forms of the corresponding Walsh matrixes are given and the fast algorithms of Walsh transforms are designed. Some important characters such as dyadic shift theorem are also introduced. 2. After the relation of code of the number and the order of Walsh functions is examined, a new kind of order of Walsh functions is proposed and its mathematic definitions and the recursion forms of the corresponding Walsh matrix are extensively studied. Similarly, the evolving generation and some relative properties such as totally parity copying character of it are discussed, some intrinsic relations between the order and the five traditional orders is also discussed. Some fast algorithms of Walsh transform based on this order are given, too.3. After the brief retrospect of spectral methods applied to get the numerical solutions of partial differential equations, the Walsh spectral methods are extensively studied. A new method of designing the "derivate operator" and a new Walsh spectral method based on quasi-Walsh order are proposed, which primarily prove the special uses of it. Numerical experiments demonstrate the efficiency of this method.