C[0,1]上完备的三角形正交函数系

A New Complete Orthogonal System for Functions in C [0,1]

基于W alsh函数,建立一个新的函数系T n n∈N.它是一个分段线性函数序列,在0,1的子区间上其函数图像呈现三角形。并且证明了它是C[0,1]上一个完备的正交函数系。在此函数系上展成的Fourier级数有许多与三角Fourier级数相似的性质,三角形Fourier级数的部分和在作为函数的逼近工具时确实要比三角Fourier级数优越一些。

Based on the Walsh functions, a new orthogonal system {Tn}n∈N is presented, which is a sequence of piecewise linear functions that are of form triangle shape on subintervals of [0, 1 ], named triangle orthogonal system. It is complete for continuous functions on [0,1] （ C[0,1 ]）, and the properties of quick transformation and other well properties similar to Fourier system, but its partial sum is superiority to Fourier system as the approaching tool.