一类频带有限的M进制正交小波基

为了将有限区间内的函数展成小波级数并克服边界的麻烦，首先给出构造频带有限的M(整数M≥2)进制尺度函数的充要条件．然后利用正交多分辨率分析，给出由该尺度函数构造的一类频带有限的M进制正交小波函数的必要条件．由该小波函数和加权多项式可以构造出有限区间上的M进制类小波函数．于是，有限区间上的函数可表示为该区间上的尺度函数和类小波函数的有限和，用于解决微分方程数值解、回归函数估计和信号处理等问题．

On M-band Band-Limited Orthogonal Wavelet Bases

The purpose of this paper is to overcome the boundary trouble when the function in a bounded interval is expanded to wavelets. First, the necessary and sufficie nt conditions of constructing the M-band band-limited scaling function are presented. Next, by the orthogonal multiresolution analysis, the ne cessary conditions of a kind of M-band band-limited orthogona l wavelet function constructed by the scaling function are proposed. By means of weighed orthogonal polynomials and the wavelet functions, the M- band band-limited orthogonal wavelet-like functions in a bounded interval ar e then established. By the above mentioned methods, the functions in a bounded i nterval can be expanded as the sum of finite items of the scaling functions and wavelet-like functions, thus the numerical analysis of partial differential equ ations, the regression function estimation and the signal processes can be easil y dealt with.