Abstract: V system is a new class of complete orthogonal system，which consists of piecewise polynomials in L₂[0，1].The V system has multiresolution property and global/local property.It also has great advantages in representing geometric models.However，the fast algorithm of V transform is difficult to design.Using Haar functions and Legendre polynomials a new class of function system，W system，is constructed in this paper，which consists of piecewise polynomials.The approximating effect by W series is the same as that by V series.In this paper，the fast algorithm of discrete W transform of degree 1 is proposed，which partially solves the difficulties in directly designing the fast algorithm of V system.

摘要： V系统是L₂[0，1]上一类新的完备正交函数系，它由分段多项式组成，具有多分辨分析特性和全局/局部性，在几何模型的正交表达方面具有明显的优势，但其快速算法难以得到。利用Haar函数和Legendre多项式构造了一类由分段k次多项式组成的函数系（文中称为W系），在该函数系上作函数逼近的效果等同于在V系统上的效果，并进一步讨论了一次离散W变换的快速算法，从而部分克服了直接对V系统设计快速算法的困难。