中文摘要：

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。

英文摘要：

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-uniform 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 GramSchmidt orthogonalization process. The orthogonality,reproducibility and convergence of the proposed functions are proved,and a specific constructive example is also given.