Abstract: To explore the effective mathematical tools(orthogonal function and orthogonal transformation)for signal analysis more deeply,a new orthogonal functions system composed of piecewise polynomials is constructed from the Legendre polynomials,and it is called generalized V-system of degree k.Equivalence relation between the generalized V-system of degree k and the V-system of degree k is proved.The discrete matrix corresponding to the generalized V-system of degree k is built,by which the generalized V-transform of degree k is constructed.It is proved that the generalized V-transform maintains almost all the properties of the V-transform,and also has a fast algorithm for degree one,which makes it superior to the V-transform as designing fast algorithms for the V-transform is difficult.The experimental results show that the time efficiency of the fast generalized V-transform is substantially increased compared with the V-transform.Key words:orthogonal functions;U-system;V-system;generalized V-system;fast algorithm 1相关工作正交