某些函数的Walsh展式与变式

WALSH EXPANSIONS AND WALSH-FOURIER TRANSFORMS FOR SOME FUNCTIONS

由于Walsh函数系是一个完整标准直交系,所以我们可如同三角函数系一样讨论可积函数的Walsh展式(W展式)。但欲具体求出一般函数的W展式并不是一件容易的事。关于Walsh变式(W变式),已经知道有与Fouricr变式相仿的一些性质,但是求函数的Waslsh变式往往更为困难。目前除掉特征函数、锯齿函数等外,能直接求出W变式的其它函数类型并不很多,本文讨论函数X^n(nN)的Walsh展式与变式。

In this paper, we focus on expanding x^n with n∈{1,2,3…}, x∈[0,1), as a Walsh-Fourier series. Moreover, for the function f(x)=x^n, x∈[0,1); 0, x∈[l,∞), we shall obtain its W-Transformation. We than show the gencral formula of the Walsh-Fourier transfomation by using the walsh expansion of x^n.