紧支撑半正交小波的构造

众所周知,一个紧支撑细分函数若生成一个正交的多分辨分析(MRA),则该MRA可容许一个具有紧支撑的小波,并且小波有明确的表达式.但若生成一个非正交的多分辨分析,我们却没有一个构造紧支撑半正交小波的一般方法.Chui与Wang曾在尺度函数紧支对称的前提下,给出了一个构造紧支撑半正交小波的方法,给出了紧支撑半正交样条小波的构造例子.但这种方法依赖于一个多项式零点的确定,在大多数情况下这是不很容易的.本文在适当的条件下,给出了一个构造半正交小波的方法.这种构造继承了来源于尺度函数和符号的对称性和消失矩性质.我们也给出了两个例子来说明这种方法.

On the Construction of Compactly Supported Prewavelets

It is well known that,if a compactly supported refinable function generates an orthogonal MRA,then the basic wavelet can be explicitly constructed and compactly supported.But interestingly if it generates a nonorthogonal MRA,there has not been a general recipe to construct a compactly supported basic prewavelet.Chui and Wang obtained an approach to construct a compactly supported basic prewavelet under the hypothesis that the scaling function is symmetric.Unfortunately,the approach by Chui and Wang highly depends on the determination of zeros of a polynomial,which is difficult in most cases.Using many properties of spline functions,Chui and Wang also constructed compactly supported spline prewavelets.In this paper,a construction of basic prewavelet is obtained under a mild hypothesis.What is more,this construction inherits the symmetry and vanishing moment originating in scaling functions and symbols.Two examples are also given to illustrate the general theory.