高次 Haar 函数的推广

摘要

The k degree V-system is a class of orthogonal piecewise polynomial functions which is also named as high order Haar functions.V-system is defined on the uniform partition of interval [0,1 ]and obtained by multi-scale squeezing and shifting operations on the so-called generators.The V-system to the case of non-uniform partition is generalized,and the corresponding result is named as high order non-uni-form Haar functions.For any given partition on the interval [0,1 ],a set of truncated monomials was firstly defined.It is proved that the non-uniform Haar functions can be obtained through the Gram-Schmidt orthogonalization process.The orthogonality,reproducibility and convergence of the proposed functions are proved,and a specific constructive example is also given.%k 次 V-系统是一类正交分段多项式函数系，Haar 函数是当 k ＝0时的情形，因而又称为高次 Haar 函数。V-系统定义在区间[0，1]上的均匀剖分上，经过对所谓“生成元函数”进行2n 倍压缩及平移得到。提出了一种正交非均匀分段多项式函数系的构造方法，称之为高次非均匀 Haar 函数系。对于任意给定的区间[0，1]上的非均匀层次嵌套剖分，首先定义一组截断单项式，并证明了对这组截断单项式系进行 Gram-Schmidt 过程，结果便是相应的高次非均匀 Haar 函数，原来的 V －系统只是高次非均匀 Haar 函数系的特殊情形。证明了该函数系的正交性，再生性及收敛性，并给出了一个具体构造实例。