a尺度二维四向双正交多小波的构造和Mallat算法

The Construction and the Mallat Algorithm of Biorthogonal Two Dimensional Four-Direction Multi-Wavelet with Dilation Factor a

摘要: 通过二维四向多加细函数以及二维四向多小波的概念，推广二尺度二维四向正交多小波为a尺度二维四向双正交多小波，更进一步对于a尺度二维四向双正交多加细函数和多小波的构造算法做出了相应给出，最后研究了a尺度二维四向多小波的Mallat算法。

Abstract: Through the concept of two dimensional four-direction multi-scaling function and two dimensional four-direction multi-wavelet, the two dimensional four-direction orthogonal multi-wavelet with dilation factor two is generalized to two dimensional biorthogonal multi-wavelet with dilation factor a. Furthermore, the construction algorithm of two dimensional biorthogonal multi-scaling function and multi-wavelet with dilation factor a is given. Finally, the Mallat algorithm of two dimensional four-direction multi-wavelet with dilation factor a is studied.

1. 引言

小波分析是这些年来发展起来的一门新兴数学理论以及方法，在信号处理，语音处理，图像处理，数据压缩，微分方程求解，地震勘探等各个领域有着广泛的应用。Haar小波是同时具有正交性，对称性和紧支撑性的单小波，但是其它单小波并不能具有这样好的性质，所以人们引入了多小波。从带宽来看，二尺度小波高频端的带宽比较窄，那么从小波分析的效果来看二尺度小波效果相对比较差，所以人们提出了a尺度小波。双向小波的概念是杨守志等人首先提出的 [1]，后来进一步得出了一系列好的理论和结果 [2] [3]。本文在引入二维四向多小波基础上，建立了a尺度二维四向多加细函数和a尺度二维四向多小波，给出了a尺度二维四向多加细函数和a尺度二维四向多小波的正交和双正交准则，以及它们的构造算法，最后讨论了a尺度二维四向多小波的分解与重构的Mallat算法。

2. 预备知识

先给出文章要提到的记号： $C^N$ 表示N维复欧几里德空间。 $I$ 表示 $r\times r$ 阶单位矩阵， $O$ 表示 $r\times r$ 阶零矩阵， $T$ 表示向量或矩阵的转置，向量值函数信号空间 $L^2\left(R\mathrm{,}{C}^N\right)\triangleq L^2$ 可以表示为 $L^2\left(R\mathrm{,}{C}^N\right)\triangleq \left\{F\left(t\right)={\left(f_1\left(t\right)\mathrm{,}{f}_2\left(t\right)\mathrm{,}\cdots \mathrm{,}{f}_N\left(t\right)\right)}^{\text{T}}\mathrm{<mo>\; :\; </mo>}t\in R\mathrm{,}{f}_k\left(t\right)\in L^2\left(R\right)\mathrm{,}k=\mathrm{1,2,}\cdots \mathrm{,}N\right\}$，(根据文献 [4] )对于 $F\left(t\right)\in L^2\left(R\mathrm{,}{C}^N\right)$ 它的积分和Fourier变换分别定义为

${\displaystyle {\int }_R}F\left(t\right)\text{d}t\triangleq {\left[{\displaystyle {\int }_R}{f}_1\left(t\right)\text{d}t\mathrm{,}{\displaystyle {\int }_R}{f}_2\left(t\right)\text{d}t\mathrm{,}\cdots \mathrm{,}{\displaystyle {\int }_R}{f}_N\left(t\right)\text{d}t\right]}^{\text{T}}$

和

$F\left(\omega \right)\triangleq {\left[{\displaystyle {\int }_R}{f}_1\left(t\right){\text{e}}^{-i\omega t}\text{d}t\mathrm{,}{\displaystyle {\int }_R}{f}_2\left(t\right){\text{e}}^{-i\omega t}\text{d}t\mathrm{,}\cdots \mathrm{,}{\displaystyle {\int }_R}{f}_N\left(t\right){\text{e}}^{-i\omega t}\text{d}t\right]}^{\text{T}}$

若F和G都是一元函数空间，它们两个的基底分别为 ${\left\{f_k\left(x\right)\right\}}_{k\in Z}$ 和 ${\left\{g_k\left(x\right)\right\}}_{k\in Z}$，二元函数空间H表示为 $H=F\otimes G$ 是F和G的张量积空间，H的基底可以表示为 ${\left\{f_k\left(x\right)g_k\left(x\right)\right\}}_{k\in Z}$。

定义 $\left\{V_j\right\}$ 的张量积空间为 $V_j^2=V_j\otimes V_j={\left\{f_k\left(x\right)g_k\left(x\right)\right\}}_{f\left(x\right)\mathrm{,}g\left(x\right)\in V_j}$，而 $\left\{V_j\right\}$ 表示为由一元尺度函数 $\phi$ 生成的一个正交多分辨分析。那么关于二元函数 $f\left(x\mathrm{,}y\right)\in L^2\left(R^2\right)$，引入记号 $\phi \left(x\mathrm{,}y\right)=\phi \left(x\right)\phi \left(y\right)$。

二维四向加细尺度函数

基于双向尺度函数的概念，现在假设有 $r+1$ 个双向尺度函数 ${\phi }_1\left(x\right)\mathrm{,}{\phi }_2\left(x\right)\mathrm{,}\cdots \mathrm{,}{\phi }_r\left(x\right),\phi \left(y\right)\in L^2\left(R\right)$，记 $\Phi \left(x\right)={\left[{\phi }_1\left(x\right)\mathrm{,}{\phi }_2\left(x\right)\mathrm{,}\cdots \mathrm{,}{\phi }_r\left(x\right)\right]}^{\text{T}}$，那么通过 $\Phi \left(x\right)$ 和 $\phi \left(y\right)$ 的张量积来构造二维四向多尺度函数。假设双向加细函数 ${\phi }_i\left(x\right)$ 和 $\phi \left(y\right)$ 都符合细分方程

${\phi }_i\left(x\right)={\displaystyle \underset{k}{\sum }}{P}_{\mathrm{1,}k}^{+}{\phi }_i\left(ax-k\right)+{\displaystyle \underset{k}{\sum }}{P}_{\mathrm{1,}k}^{-}{\phi }_i\left(k-ax\right)\mathrm{,}i=\mathrm{1,}\cdots \mathrm{,}r$

$\phi \left(y\right)={\displaystyle \underset{k}{\sum }}{P}_{\mathrm{2,}k}^{+}\phi \left(ay-k\right)+{\displaystyle \underset{k}{\sum }}{P}_{\mathrm{2,}k}^{-}\phi \left(k-ay\right)$

设 $\Phi \left(x\mathrm{,}y\right)={\phi }_i\left(x\right)\phi \left(y\right)$，就能得出

$\begin{array}{c}\Phi \left(x\mathrm{,}y\right)=\left\{{\displaystyle \underset{k}{\sum }}{P}_{\mathrm{1,}k}^{+}{\phi }_i\left(ax-k\right)+{\displaystyle \underset{k}{\sum }}{P}_{\mathrm{1,}k}^{-}{\phi }_i\left(k-ax\right)\right\}\otimes \left\{{\displaystyle \underset{l}{\sum }}{P}_{\mathrm{2,}l}^{+}\phi \left(ay-l\right)+{\displaystyle \underset{l}{\sum }}{P}_{\mathrm{2,}l}^{-}\phi \left(l-ay\right)\right\}\\ ={\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{\mathrm{1,}k}^{+}{P}_{\mathrm{2,}l}^{+}{\phi }_i\left(ax-k\right)\phi \left(ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{\mathrm{1,}k}^{+}{P}_{\mathrm{2,}l}^{-}{\phi }_i\left(ax-k\right)\phi \left(l-ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{\mathrm{1,}k}^{-}{P}_{\mathrm{2,}l}^{+}{\phi }_i\left(k-ax\right)\phi \left(ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{\mathrm{1,}k}^{-}{P}_{\mathrm{2,}l}^{-}{\phi }_i\left(k-ax\right)\phi \left(l-ay\right)\\ ={\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{\mathrm{1,}k}^{+}{P}_{\mathrm{2,}l}^{+}\phi \left(ax-k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{\mathrm{1,}k}^{+}{P}_{\mathrm{2,}l}^{-}\phi \left(ax-k\mathrm{,}l-ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{\mathrm{1,}k}^{-}{P}_{\mathrm{2,}l}^{+}\phi \left(k-ax\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{\mathrm{1,}k}^{-}{P}_{\mathrm{2,}l}^{-}\phi \left(k-ax\mathrm{,}l-ay\right).\end{array}$

从而根据适合的 $\left\{P_{l\mathrm{,}k}^{+\mathrm{,}+}\right\}\mathrm{,}\left\{P_{l\mathrm{,}k}^{+\mathrm{,}-}\right\}\mathrm{,}\left\{P_{l\mathrm{,}k}^{-\mathrm{,}+}\right\}\mathrm{,}\left\{P_{l\mathrm{,}k}^{-\mathrm{,}-}\right\}$，就有

$\begin{array}{c}\Phi \left(x\mathrm{,}y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}+}\phi \left(ax-k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}-}\phi \left(ax-k\mathrm{,}l-ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}+}\phi \left(k-ax\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}-}\phi \left(k-ax\mathrm{,}l-ay\right)\mathrm{.}\end{array}$

令 $\Phi \left(x\mathrm{,}y\right)=\Phi \left(x\right)\cdot \phi \left(y\right)$，则

$\Phi \left(x,y\right)={\left[{\Phi }_1\left(x,y\right),{\Phi }_2\left(x,y\right),\cdots ,{\Phi }_r\left(x,y\right)\right]}^{\text{T}}$

那么就可以有a尺度多小波细分方程

$\begin{array}{c}\Phi \left(x\mathrm{,}y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}+}\Phi \left(ax-k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}-}\Phi \left(ax-k\mathrm{,}l-ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}+}\Phi \left(k-ax\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}-}\Phi \left(k-ax\mathrm{,}l-ay\right)\mathrm{.}\end{array}$ (1)

接下来对(1)式进行Fourier变换就可以有

$\begin{array}{c}\hat{\Phi }\left({\omega }_1\mathrm{,}{\omega }_2\right)=P^{+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)+P^{+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\\ +P^{-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)+P^{-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\mathrm{,}\end{array}$ (2)

其中

$P^{+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{k\mathrm{,}l}^{+\mathrm{,}+}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}$

$P^{+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{k\mathrm{,}l}^{+\mathrm{,}-}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}$

$P^{-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{k\mathrm{,}l}^{-\mathrm{,}+}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}$

$P^{-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{k\mathrm{,}l}^{-\mathrm{,}-}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}$

令

$P_{k\mathrm{,}l}^{-\mathrm{,}+}=\left[\begin{array}{cc}{P}_{\mathrm{1,}k\mathrm{,}l}^{-\mathrm{,}+}& P_{r\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\\ 0& P_{r^2\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\end{array}\right]\mathrm{,}{P}_{k\mathrm{,}l}^{-\mathrm{,}-}=\left[\begin{array}{cc}{P}_{\mathrm{1,}k\mathrm{,}l}^{-\mathrm{,}-}& P_{r\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\\ 0& P_{r^2\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\end{array}\right]\mathrm{,}$

为双正，正负和双负矩阵符号。下面对(1)进行变形有

$\begin{array}{c}\Phi \left(x,-y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}+}\Phi \left(x\right)\left(ax-k\mathrm{,}-ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}-}\Phi \left(x\right)\left(ax-k\mathrm{,}l+ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}+}\Phi \left(x\right)\left(k-ax\mathrm{,}-ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}-}\Phi \left(x\right)\left(k-ax\mathrm{,}l+ay\right)\mathrm{.}\end{array}$ (3)

$\begin{array}{c}\Phi \left(-x,y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}+}\Phi \left(x\right)\left(-ax-k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}-}\Phi \left(x\right)\left(-ax-k\mathrm{,}l-ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}+}\Phi \left(x\right)\left(ax+k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}-}\Phi \left(x\right)\left(ax+k\mathrm{,}l-ay\right)\mathrm{.}\end{array}$ (4)

$\begin{array}{c}\Phi \left(-x,-y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}+}\Phi \left(x\right)\left(-ax-k\mathrm{,}-ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}-}\Phi \left(x\right)\left(-ax-k\mathrm{,}l+ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}+}\Phi \left(x\right)\left(ax+k\mathrm{,}-ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}-}\Phi \left(x\right)\left(ax+k\mathrm{,}l+ay\right)\mathrm{.}\end{array}$ (5)

下面对(3)~(5)式都进行Fourier变换就有

$\begin{array}{c}\hat{\Phi }\left({\omega }_1\mathrm{,}-{\omega }_2\right)=P^{+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)+P^{+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\\ +P^{-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)+P^{-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\mathrm{,}\end{array}$ (6)

$\begin{array}{c}\hat{\Phi }\left(-{\omega }_1\mathrm{,}{\omega }_2\right)=P^{+\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)+P^{+\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\\ +P^{-\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)+P^{-\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\mathrm{,}\end{array}$ (7)

$\begin{array}{c}\hat{\Phi }\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)=P^{+\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)+P^{+\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\\ +P^{-\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)+P^{-\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\mathrm{,}\end{array}$ (8)

根据(2)(6)~(8)式可以有，令

$\Gamma \left(x\mathrm{,}y\right)={\left[\Phi \left(x\mathrm{,}y\right)\mathrm{,}\Phi \left(x\mathrm{,}-y\right)\mathrm{,}\Phi \left(-x\mathrm{,}y\right)\mathrm{,}\Phi \left(-x\mathrm{,}-y\right)\right]}^{\text{T}}\mathrm{,}$

$\hat{\Gamma }\left({\omega }_1\mathrm{,}{\omega }_2\right)=\left[\begin{array}{cccc}{P}^{+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\\ P^{+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\\ P^{-\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\\ P^{-\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\end{array}\right]\hat{\Gamma }\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\mathrm{,}$ (9)

方程(2)有解，当且仅当式(9)有解。

设

$\Gamma \left(x\mathrm{,}y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}\left[\begin{array}{cccc}{P}_{k\mathrm{,}l}^{+\mathrm{,}+}& P_{k\mathrm{,}l}^{+\mathrm{,}-}& P_{k\mathrm{,}l}^{-\mathrm{,}+}& P_{k\mathrm{,}l}^{-\mathrm{,}-}\\ P_{k\mathrm{,}-l}^{+\mathrm{,}-}& P_{k\mathrm{,}-l}^{+\mathrm{,}+}& P_{k\mathrm{,}-l}^{-\mathrm{,}-}& P_{k\mathrm{,}-l}^{-\mathrm{,}+}\\ P_{-k\mathrm{,}l}^{-\mathrm{,}+}& P_{-k\mathrm{,}l}^{-\mathrm{,}-}& P_{-k\mathrm{,}l}^{+\mathrm{,}+}& P_{-k\mathrm{,}l}^{+\mathrm{,}-}\\ P_{-k\mathrm{,}-l}^{-\mathrm{,}-}& P_{-k\mathrm{,}-l}^{-\mathrm{,}+}& P_{-k\mathrm{,}-l}^{+\mathrm{,}-}& P_{-k\mathrm{,}-l}^{+\mathrm{,}+}\end{array}\right]\Gamma \left(ax-k\mathrm{,}ay-l\right)\mathrm{,}$ (10)

则(9)式为 $\Gamma \left(x\mathrm{,}y\right)$ 在频域里的a尺度加密方程，它的加细面具符号为

$P\left({\omega }_1\mathrm{,}{\omega }_2\right)=\left[\begin{array}{cccc}{P}^{+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\\ P^{+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\\ P^{-\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\\ P^{-\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}\text{}-\frac{{\omega }_2}{a}\right)& P^{-\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}\text{}-\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}-}\left(-\frac{{\omega }_1}{a}\mathrm{,}\text{}-\frac{{\omega }_2}{a}\right)& P^{+\mathrm{,}+}\left(-\frac{{\omega }_1}{a}\mathrm{,}\text{}-\frac{{\omega }_2}{a}\right)\end{array}\right]$ (11)

定义方程(1)的自相关矩阵符号

$\Omega \left(x\mathrm{,}y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}\left[\begin{array}{cccc}{\Omega }_{11}& {\Omega }_{12}& {\Omega }_{13}& {\Omega }_{14}\\ {\Omega }_{21}& {\Omega }_{22}& {\Omega }_{23}& {\Omega }_{24}\\ {\Omega }_{31}& {\Omega }_{32}& {\Omega }_{33}& {\Omega }_{34}\\ {\Omega }_{41}& {\Omega }_{42}& {\Omega }_{43}& {\Omega }_{44}\end{array}\right]{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}$

其中

${\Omega }_{11}=\langle \Phi \left(x,y\right),\Phi \left(x-k,y-l\right)\rangle ;{\Omega }_{12}=\langle \Phi \left(x,y\right),\Phi \left(x-k,l-y\right)\rangle ;$

${\Omega }_{13}=\langle \Phi \left(x,y\right),\Phi \left(k-x,y-l\right)\rangle ;{\Omega }_{14}=\langle \Phi \left(x,y\right),\Phi \left(k-x,l-y\right)\rangle ;$

${\Omega }_{21}=\langle \Phi \left(x,-y\right),\Phi \left(x-k,y-l\right)\rangle ;{\Omega }_{22}=\langle \Phi \left(x,-y\right),\Phi \left(x-k,l-y\right)\rangle ;$

${\Omega }_{23}=\langle \Phi \left(x,-y\right),\Phi \left(k-x,y-l\right)\rangle ;{\Omega }_{24}=\langle \Phi \left(x,-y\right),\Phi \left(k-x,l-y\right)\rangle ;$

${\Omega }_{31}=\langle \Phi \left(-x,y\right),\Phi \left(x-k,y-l\right)\rangle ;{\Omega }_{32}=\langle \Phi \left(-x,y\right),\Phi \left(x-k,l-y\right)\rangle ;$

${\Omega }_{33}=\langle \Phi \left(-x,y\right),\Phi \left(k-x,y-l\right)\rangle ;{\Omega }_{34}=\langle \Phi \left(-x,y\right),\Phi \left(k-x,l-y\right)\rangle ;$

${\Omega }_{41}=\langle \Phi \left(-x,-y\right),\Phi \left(x-k,y-l\right)\rangle ;{\Omega }_{42}=\langle \Phi \left(-x,-y\right),\Phi \left(x-k,l-y\right)\rangle ;$

${\Omega }_{43}=\langle \Phi \left(-x,-y\right),\Phi \left(k-x,y-l\right)\rangle ;{\Omega }_{44}=\langle \Phi \left(-x,-y\right),\Phi \left(k-x,l-y\right)\rangle .$

下面引入变换算子 $\tau$ ：

$\tau A\left({\left({\omega }_1\right)}^a\mathrm{,}{\left({\omega }_2\right)}^a\right)={\displaystyle \underset{k\mathrm{,}l=0}{\overset{a-1}{\sum }}}P\left({\omega }_1+\frac{2k\pi }{a}\mathrm{,}{\omega }_2+\frac{2l\pi }{a}\right)A\left({\omega }_1+\frac{2k\pi }{a}\mathrm{,}{\omega }_2+\frac{2l\pi }{a}\right)P^{\mathrm{<mo>\; *\; </mo>}}\left({\omega }_1+\frac{2k\pi }{a}\mathrm{,}{\omega }_2+\frac{2l\pi }{a}\right)\mathrm{,}$ (12)

其中 $A\left({\left({\omega }_1\right)}^a\mathrm{,}{\left({\omega }_2\right)}^a\right)$ 是 $P\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 的Laurent多项式， $P\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 由(12)式可知。

关于 $\Omega \left({\omega }_1\mathrm{,}{\omega }_2\right)$ 和 $\tau$ 可以有下面的引理。

引理1：矩阵符号 $\Omega \left({\omega }_1\mathrm{,}{\omega }_2\right)$ 和变换算子 $\tau$ 的定义如上，那么由Poisson求和公式可得

$\Omega \left({\omega }_1\mathrm{,}{\omega }_2\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}\left[\begin{array}{c}\Phi \left({\omega }_1+\frac{2k\pi }{a}\mathrm{,}{\omega }_2+\frac{2l\pi }{a}\right)\\ \Phi \left({\omega }_1+\frac{2k\pi }{a}\mathrm{,}-{\omega }_2-\frac{2l\pi }{a}\right)\\ \Phi \left(-{\omega }_1-\frac{2k\pi }{a}\mathrm{,}{\omega }_2+\frac{2l\pi }{a}\right)\\ \Phi \left(-{\omega }_1-\frac{2k\pi }{a}\mathrm{,}-{\omega }_2-\frac{2l\pi }{a}\right)\end{array}\right]{\left[\begin{array}{c}\Phi \left({\omega }_1+\frac{2k\pi }{a}\mathrm{,}{\omega }_2+\frac{2l\pi }{a}\right)\\ \Phi \left({\omega }_1+\frac{2k\pi }{a}\mathrm{,}-{\omega }_2-\frac{2l\pi }{a}\right)\\ \Phi \left(-{\omega }_1-\frac{2k\pi }{a}\mathrm{,}{\omega }_2+\frac{2l\pi }{a}\right)\\ \Phi \left(-{\omega }_1-\frac{2k\pi }{a}\mathrm{,}-{\omega }_2-\frac{2l\pi }{a}\right)\end{array}\right]}^{\text{T}}$

进一步，我们可以得出 $\Omega \left({\omega }_1\mathrm{,}{\omega }_2\right)$ 是 $\tau$ 相应于特征值为1的特征矩阵。

定理1：加细方程(1)有紧支撑解当且仅当它的面具符号满足下面情况之一

1) $(\begin{array}{l}{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)=a^{2}I\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{<mo>\; ;\; </mo>}\hfill \end{array}$

2) $(\begin{array}{l}\left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ {\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)=a^{2}I\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{<mo>\; ;\; </mo>}\hfill \end{array}$

3) $(\begin{array}{l}\left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ {\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)=a^{2}I\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{<mo>\; ;\; </mo>}\hfill \end{array}$

4) $(\begin{array}{l}\left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ \left|{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)\right|\le a^2\mathrm{,}\hfill \\ {\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)=a^{2}I\mathrm{<mo>\; ;\; </mo>}\hfill \end{array}$

证明：根据文献 [5] [6] [7] [8]，方程(10)存在紧支撑分布解当且仅当1是(11)式定义矩阵 $P\left(\mathrm{1,1}\right)$ 的一个特征值， $P\left(\mathrm{1,1}\right)$ 的其他特征值的模都不大于1。另外， $P\left(\mathrm{1,1}\right)$ 的4个特征值分别是， $\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}+P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)$， $\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}+P_{k\mathrm{,}l}^{+\mathrm{,}-}-P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)$ 以及 $\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}\left(P_{k\mathrm{,}l}^{+\mathrm{,}+}-P_{k\mathrm{,}l}^{+\mathrm{,}-}+P_{k\mathrm{,}l}^{-\mathrm{,}+}-P_{k\mathrm{,}l}^{-\mathrm{,}-}\right)$。定理易证。

定理2：若要 $\Phi \left(x\mathrm{,}y\right)$ 有紧支撑性，则要证明每个分量是紧支撑的。设 $\Phi \left(x\right)={\left[{\phi }_1\left(x\right)\mathrm{,}{\phi }_2\left(x\right)\mathrm{,}\cdots \mathrm{,}{\phi }_r\left(x\right)\right]}^{\text{T}}$，其中 ${\phi }_i\left(x\right)$ 和 $\phi \left(y\right)$ 是双向细分函数满足

${\phi }_i\left(x\right)={\displaystyle \underset{k=0}{\overset{N_1}{\sum }}}{P}_k^{+}{\phi }_i\left(ax-k\right)+{\displaystyle \underset{k=-N_1}{\overset{0}{\sum }}}{P}_k^{-}{\phi }_i\left(k-ax\right)$

$\phi \left(y\right)={\displaystyle \underset{k=0}{\overset{N_2}{\sum }}}{p}_k^{+}\phi \left(ay-k\right)+{\displaystyle \underset{k=-N_2}{\overset{0}{\sum }}}{p}_k^{-}\phi \left(k-ay\right)$

如果 $\Phi \left(x\right)$ 和 $\phi \left(y\right)$ 是紧支撑的，那么可根据 $\Phi \left(x\right)$ 和 $\phi \left(y\right)$ 的张量积生成二维四向加细函数 $Supp\Phi \left(x\right)\subseteq \left[-\frac{N_1}{a-1}\mathrm{,}\frac{N_1}{a-1}\right]$， $Supp\phi \left(y\right)\subseteq \left[-\frac{N_2}{a-1}\mathrm{,}\frac{N_2}{a-1}\right]$，则有

$Supp\Phi \left(x\mathrm{,}y\right)=\Phi \left(x\right)\cdot \phi \left(y\right)\subseteq \left[-\frac{N_1}{a-1}\mathrm{,}\frac{N_1}{a-1}\right]\times \left[-\frac{N_2}{a-1}\mathrm{,}\frac{N_2}{a-1}\right]$

证明：根据文献 [9] 中定理4可知。

3. a尺度二维四向多分辨分析

定义子空间序列 ${\left\{V_j\right\}}_{j\in Z}\subset L^2\left(R^2\right)$，

$\begin{array}{l}{V}_j=Clos{e}_{L^2\left(R^2\right)}\langle a^j\Phi \left(a^{j}x-k\mathrm{,}{a}^{j}y-l\right)\mathrm{,}{a}^j\Phi \left(a^{j}x-k\mathrm{,}l-a^{j}y\right)\mathrm{,}\\ a^j\Phi \left(k-a^{j}x\mathrm{,}{a}^{j}y-l\right)\mathrm{,}{a}^j\Phi \left(k-a^{j}x\mathrm{,}l-a^{j}y\right)\mathrm{,}k\mathrm{,}l\in Z\rangle \mathrm{,}\end{array}$ (13)

那么要产生在 $L^2\left(R^2\right)$ 中的一个多分辨分析 ${\left\{V_j\right\}}_{j\in Z}$ 当且仅当(13)式里的 $V_j$ 应当满足以下条件：

1) $\cdots \subset V_{-1}\subset V_0\subset V_1\subset \cdots$ ；

2) $Clos{e}_{L^2\left(R^2\right)}{\displaystyle {\cup }_{j\in Z}}{V}_j=L^2\left(R^2\right)$ ；

3) ${\displaystyle {\cap }_{j\in Z}}{V}_j=\left\{0\right\}$ ；

4) $f\left(x\mathrm{,}y\right)\in V_j\iff f\left(ax\mathrm{,}ay\right)\in V_{j+1}$ ；

5) 存在 $L^2\left(R^2\right)$ 里的一个函数 $\Phi \left(x\mathrm{,}y\right)$，使得集合 $\left\{\Phi \left(x-k\mathrm{,}y-l\right)\mathrm{,}\Phi \left(x-k\mathrm{,}l-y\right)\mathrm{,}\Phi \left(k-x\mathrm{,}y-l\right)\mathrm{,}\Phi \left(k-x\mathrm{,}l-y\right)\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\right\}$ 是 $V_0$ 的Riesz基，那么就有两个常数 $0<A\le B<\infty$，则对于系数向量序列

${\left\{c^{k\mathrm{,}l}\right\}}_{k\mathrm{,}l\in Z}={\left\{\left[c_1^{k\mathrm{,}l}\mathrm{,}{c}_2^{k\mathrm{,}l}\mathrm{,}{c}_3^{k\mathrm{,}l}\mathrm{,}{c}_4^{k\mathrm{,}l}\right]\right\}}_{k\mathrm{,}l\in Z}\subset l^2\left(Z^4\right)$ 有

$\begin{array}{c}A{\Vert {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_{k\mathrm{,}l}{c}_{k\mathrm{,}l}^{\mathrm{<mo>\; *\; </mo>}}\Vert }_2^2\le \Vert {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_1^{k\mathrm{,}l}\Phi \left(x-k\mathrm{,}y-l\right)+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_2^{k\mathrm{,}l}\Phi \left(x-k\mathrm{,}l-y\right)+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_3^{k\mathrm{,}l}\Phi \left(k-x\mathrm{,}y-l\right)+{{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_4^{k\mathrm{,}l}\Phi \left(k-x\mathrm{,}l-y\right)\Vert }_2^2\\ \le B{\Vert {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_{k\mathrm{,}l}{c}_{k\mathrm{,}l}^{\mathrm{<mo>\; *\; </mo>}}\Vert }_2^2\mathrm{,}\end{array}$ (14)

称式(14)为稳定性条件。

根据多重多分辨分析的性质 $f\left(x\mathrm{,}y\right)\in V_j\iff f\left(x+\frac{k}{a^j}\mathrm{,}y+\frac{l}{a^j}\right)\in V_j$，可以定义：

${\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}=a^j\Phi \left(a^{j}x-k\mathrm{,}{a}^{j}y-l\right)\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}=a^j\Phi \left(a^{j}x-k\mathrm{,}l-a^{j}y\right)\mathrm{,}$

${\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}=a^j\Phi \left(k-a^{j}x\mathrm{,}{a}^{j}y-l\right)\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}=a^j\Phi \left(k-a^{j}x\mathrm{,}l-a^{j}y\right)\mathrm{.}$

则 $\left\{{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\right\}$ 同样可以构成 $V_j$ 的Riesz基，有

$A{\Vert {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_{k\mathrm{,}l}{c}_{k\mathrm{,}l}^{\mathrm{<mo>\; *\; </mo>}}\Vert }_2^2\le {\Vert {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_1^{k\mathrm{,}l}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_2^{k\mathrm{,}l}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_3^{k\mathrm{,}l}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_4^{k\mathrm{,}l}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\Vert }_2^2\le B{\Vert {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{c}_{k\mathrm{,}l}{c}_{k\mathrm{,}l}^{\mathrm{<mo>\; *\; </mo>}}\Vert }_2^2\mathrm{,}$

因为 $\Phi \left(x\mathrm{,}y\right)\in V_0\subset V_1$，并且 $\left\{{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\right\}$ 同样也可以构成 $V_1$ 的Riesz基，故有 $c_1^{k\mathrm{,}l}\mathrm{,}{c}_2^{k\mathrm{,}l}\mathrm{,}{c}_3^{k\mathrm{,}l}\mathrm{,}{c}_4^{k\mathrm{,}l}\in l^2\left(Z^4\right)$，从而使 $\Phi \left(x\mathrm{,}y\right)$ 满足(1)式。

定理3：如果尺度函数 $\Phi \left(x\mathrm{,}y\right)\in L^2\left(R^2\right)$ 符合多分辨分析，现定义 $V_j=\left\{{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\right\}$ 构成 $V_0$ 的Riesz基。若存在函数集 $\left\{\Phi \left(x-k\mathrm{,}y-l\right)\mathrm{,}\Phi \left(x-k\mathrm{,}l-y\right)\mathrm{,}\Phi \left(k-x\mathrm{,}y-l\right)\mathrm{,}\Phi \left(k-x\mathrm{,}l-y\right)\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\right\}$ 构成 $V_0$ 的Riesz基，那么 ${\displaystyle {\cap }_{j\in Z}}{V}_j=\left\{0\right\}$。

证明：由于 $\left\{{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\right\}$ 构成 $V_0$ 的Riesz基，故存在两个常数 $0<A\le B<\infty$，对任意的 $f\left(x\mathrm{,}y\right)\in V_0$，有

$A{\Vert f\Vert }_2^2\le {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{-\mathrm{,}-}\rangle \right|}^2\le B{\Vert f\Vert }_2^2\mathrm{,}$

所以对于所有的 $f\left(x\mathrm{,}y\right)\in V_j$，就有

$A{\Vert f\Vert }_2^2\le {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\rangle \right|}^2\le B{\Vert f\Vert }_2^2\mathrm{,}$

对于 $\forall \epsilon >0$， $f\left(x\mathrm{,}y\right)\in {\displaystyle {\cap }_{j\in Z}}{V}_j$，存在一个紧支撑连续函数 $\tilde{f}\left(x\mathrm{,}y\right)$，使 ${\Vert f-\tilde{f}\Vert }_{L^2}\le \epsilon$。

若 $P_j$ 是 $V_j$ 的正交投影算子，则 $\Vert f\Vert -\Vert P_j\tilde{f}\Vert \le \Vert f-P_j\tilde{f}\Vert =\Vert P_{j}f-P_j\tilde{f}\Vert \le \Vert f-\tilde{f}\Vert \le \epsilon$，故 $\Vert f\Vert \le \epsilon +\Vert P_j\tilde{f}\Vert ,\forall j\in Z$。

设 $Supp\tilde{f}\left(x\mathrm{,}y\right)=\left[-r\mathrm{,}r\right]\times \left[-r\mathrm{,}r\right]=S$，则

$\begin{array}{l}{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle \tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2\le a^{2j}{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left[{\displaystyle {\int }_{\left(x\mathrm{,}y\right)\in S}}\left|\tilde{f}\left(x\mathrm{,}y\right)\right|\left|\Phi \left(a^{j}x-k\mathrm{,}{a}^{j}y-l\right)\right|\text{d}x\text{d}y\right]}^2\\ \le a^{2j}{\Vert \tilde{f}\left(x\mathrm{,}y\right)\Vert }_{L^{\infty }}^2{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left[{\displaystyle {\int }_{\left(x\mathrm{,}y\right)\in S}}\left|\Phi \left(a^{j}x-k\mathrm{,}{a}^{j}y-l\right)\right|\text{d}x\text{d}y\right]}^2\\ \le a^{2j}{\Vert \tilde{f}\left(x\mathrm{,}y\right)\Vert }_{L^{\infty }}^2{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\displaystyle {\int }_{\left(x\mathrm{,}y\right)\in S}}{\left|\Phi \left(a^{j}x-k\mathrm{,}{a}^{j}y-l\right)\right|}^2\text{d}x\text{d}y\\ ={\Vert \tilde{f}\left(x\mathrm{,}y\right)\Vert }_{L^{\infty }}^2{\displaystyle {\int }_{D^j}}{\left|\Phi \left(x\mathrm{,}y\right)\right|}^2\text{d}x\text{d}y\end{array}$

其中 $D^j={\displaystyle \underset{k\mathrm{,}l}{\cup }\left[k-a^{j}r\mathrm{,}k+a^{j}r\right]\times \left[l-a^{j}r\mathrm{,}l+a^{j}r\right]}$。所以，当 $j\to -\infty$ 时 ${\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle \tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2\to 0$。同理，可以证明： ${\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle \tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2\to 0$， ${\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle \tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2\to 0$， ${\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle \tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\rangle \right|}^2\to 0$。又由于

${\Vert P_j\tilde{f}\Vert }^2\le \frac{1}{A}{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}\left[{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2+{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2+{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2+{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{-\mathrm{,}-}\rangle \right|}^2\right]$

所以 $\underset{j\to -\infty }{{\displaystyle \lim }}\Vert P_j\tilde{f}\Vert =0$。根据 $\Vert f\Vert \le \epsilon +\Vert P_j\tilde{f}\Vert ,\forall j\in Z$，可以知道 $f\left(x\mathrm{,}y\right)=0$。

定理4：若 $\Phi \left(x\mathrm{,}y\right)\in L^2\left(R^2\right)$ 符合定义式(1)，根据(13)式定义的 $V_j$，若 $\Phi \left(x\mathrm{,}y\right)$ 满足1) 集合 $\left\{\Phi \left(x-k\mathrm{,}y-l\right)\mathrm{,}\Phi \left(x-k\mathrm{,}l-y\right)\mathrm{,}\Phi \left(k-x\mathrm{,}y-l\right)\mathrm{,}\Phi \left(k-x\mathrm{,}l-y\right)\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\right\}$ 是 $V_0$ 的Riesz基；2) 对所有的 $\left({\omega }_1\mathrm{,}{\omega }_2\right)\in R^2$， $\hat{\Phi }\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 有界；3) $\hat{\Phi }\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 在 $\left({\omega }_1\mathrm{,}{\omega }_2\right)=\left(\mathrm{0,0}\right)$ 附近连续， $\hat{\Phi }\left({\omega }_1\mathrm{,}{\omega }_2\right)\ne 0$，那么 $Clos{e}_{L^2\left(R^2\right)}{\displaystyle {\cup }_{j\in Z}}{V}_j=L^2\left(R^2\right)$。

证明：由于 $\left\{{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\right\}$ 构成 $V_0$ 的Riesz基，故存在两个常数 $0<A\le B<\infty$，对任意的 $f\left(x\mathrm{,}y\right)\in V_0$，有

$A{\Vert f\Vert }_2^2\le {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{\mathrm{0,}k\mathrm{,}l}^{-\mathrm{,}-}\rangle \right|}^2\le B{\Vert f\Vert }_2^2\mathrm{,}$

所以对于所有的 $f\left(x\mathrm{,}y\right)\in V_j$，就有

$A{\Vert f\Vert }_2^2\le {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle f\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\rangle \right|}^2\le B{\Vert f\Vert }_2^2\mathrm{,}$

对于， $f\left(x\mathrm{,}y\right)\in {\left({\displaystyle {\cup }_{j\in Z}}{V}_j\right)}^{\perp }$，那么就有紧支撑连续函数 $\tilde{f}\left(x\mathrm{,}y\right)$，使 ${\Vert f-\tilde{f}\Vert }_{L^2}\le \epsilon$。

若 $P_j$ 是 $V_j$ 的正交投影算子，则 $\Vert P_j\tilde{f}\Vert ={\Vert P_j\left(\tilde{f}-f\right)\Vert }_{L^2}\le \epsilon$。又因为 $P_j\tilde{f}\in V_j$，则

$B{\Vert P_j\tilde{f}\Vert }^2\ge {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-,-}\rangle \right|}^2\mathrm{,}$

进一步可以得到

$\begin{array}{l}{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-,-}\rangle \right|}^2\\ =\frac{1}{{\left(2\pi \right)}^4}{\displaystyle {\int }_{-\infty }^{+\infty }}{\displaystyle {\int }_{-\infty }^{+\infty }}({\left|\hat{\Phi }\left(a^{-j}{\omega }_1\mathrm{,}{a}^{-j}{\omega }_2\right)\right|}^2+{\left|\hat{\Phi }\left(a^{-j}{\omega }_1\mathrm{,}-a^{-j}{\omega }_2\right)\right|}^2+{\left|\hat{\Phi }\left(-a^{-j}{\omega }_1\mathrm{,}{a}^{-j}{\omega }_2\right)\right|}^2\\ +{\left|\hat{\Phi }\left(-a^{-j}{\omega }_1\mathrm{,}-a^{-j}{\omega }_2\right)\right|}^2)\widehat{P_j\stackrel{˜}{f}}\left({\omega }_1\mathrm{,}{\omega }_2\right)\text{d}{\omega }_1\text{d}{\omega }_2+D\end{array}$

其中

$D\le \left({\Vert \hat{\Phi }\left({\omega }_1\mathrm{,}{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left({\omega }_1\mathrm{,}-{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left(-{\omega }_1\mathrm{,}{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)\Vert }_{L^{\infty }}^2\right)\cdot {\displaystyle \underset{k\ne 0,l\ne 0}{\sum }}{\left|P_j\tilde{f}\left(a^{j}k\mathrm{,}{a}^{j}l\right)\right|}^2\mathrm{.}$

因为 $\tilde{f}\left(x\mathrm{,}y\right)\in C^{\infty }$，所以存在一个常数M，使 $\left|\tilde{f}\left(x\mathrm{,}y\right)\right|\le M{\left(1+\left|x+y\right|\right)}^{-2}$ 故

$\begin{array}{c}D\le M^2\left({\Vert \hat{\Phi }\left({\omega }_1\mathrm{,}{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left({\omega }_1\mathrm{,}-{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left(-{\omega }_1\mathrm{,}{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)\Vert }_{L^{\infty }}^2\right)\cdot {\displaystyle \underset{k\ne 0,l\ne 0}{\sum }}{\left|P_j\tilde{f}\left(a^{j}k\mathrm{,}{a}^{j}l\right)\right|}^2\\ \le M^2\left({\Vert \hat{\Phi }\left({\omega }_1\mathrm{,}{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left({\omega }_1\mathrm{,}-{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left(-{\omega }_1\mathrm{,}{\omega }_2\right)\Vert }_{L^{\infty }}^2+{\Vert \hat{\Phi }\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)\Vert }_{L^{\infty }}^2\right)\cdot {\displaystyle \underset{k\ne 0,l\ne 0}{\sum }}{\left(1+\left|a^{j}k+a^{j}l\right|\right)}^{-2}\\ \le {M^{\prime }}^2{a}^{-2j}\end{array}$

由以上的推导，可以得到

$\begin{array}{l}\frac{1}{{\left(2\pi \right)}^4}{\displaystyle {\int }_{-\infty }^{+\infty }}{\displaystyle {\int }_{-\infty }^{+\infty }}({\left|\hat{\Phi }\left(a^{-j}{\omega }_1\mathrm{,}{a}^{-j}{\omega }_2\right)\right|}^2+{\left|\hat{\Phi }\left(a^{-j}{\omega }_1\mathrm{,}-a^{-j}{\omega }_2\right)\right|}^2+{\left|\hat{\Phi }\left(-a^{-j}{\omega }_1\mathrm{,}{a}^{-j}{\omega }_2\right)\right|}^2\\ +{\left|\hat{\Phi }\left(-a^{-j}{\omega }_1\mathrm{,}-a^{-j}{\omega }_2\right)\right|}^2)\widehat{P_j\stackrel{˜}{f}}\left({\omega }_1\mathrm{,}{\omega }_2\right)\text{d}{\omega }_1\text{d}{\omega }_2\le {\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}+}\rangle \right|}^2\\ +{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{+\mathrm{,}-}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}+}\rangle \right|}^2+{\displaystyle \underset{k\mathrm{,}l\in Z}{\sum }}{\left|\langle P_j\tilde{f}\mathrm{,}{\Phi }_{j\mathrm{,}k\mathrm{,}l}^{-\mathrm{,}-}\rangle \right|}^2+\left|D\right|\\ \le B{\epsilon }^2+{M^{\prime }}^2{a}^{-2j}\mathrm{.}\end{array}$

由于 $\Phi \left({\omega }_1\mathrm{,}{\omega }_2\right)$ 有界且在 $\left({\omega }_1\mathrm{,}{\omega }_2\right)=\left(\mathrm{0,0}\right)$ 附近处连续且 $\Phi \left(\mathrm{0,0}\right)\ne 0$，故当 $j\to +\infty$ 时，以上不等式收敛于

$\frac{1}{{\left(2\pi \right)}^4}\left({\left|\hat{\Phi }\left(\mathrm{0,0}\right)\right|}^2+{\left|\hat{\Phi }\left(\mathrm{0,0}\right)\right|}^2+{\left|\hat{\Phi }\left(\mathrm{0,0}\right)\right|}^2+{\left|\hat{\Phi }\left(\mathrm{0,0}\right)\right|}^2\right)\cdot {\Vert \widehat{P_j\stackrel{˜}{f}}\Vert }_{L^2}^2=\frac{1}{4{\pi }^4}{\left|\hat{\Phi }\left(\mathrm{0,0}\right)\right|}^2\cdot {\Vert \widehat{P_j\stackrel{˜}{f}}\Vert }_{L^2}^2$

从而 ${\Vert \widehat{P_j\stackrel{˜}{f}}\Vert }_{L^2}^2\le \frac{{\left(2\pi \right)}^{4}B{\epsilon }^2}{4{\left|\hat{\Phi }\left(\mathrm{0,0}\right)\right|}^2}=\frac{4{\pi }^{4}B{\epsilon }^2}{{\left|\hat{\Phi }\left(\mathrm{0,0}\right)\right|}^2}$，所以 ${\Vert f\Vert }_{L^2}\le \epsilon +{\Vert \widehat{P_j\stackrel{˜}{f}}\Vert }_{L^2}\le \epsilon +\frac{2{\pi }^2\sqrt{B}\epsilon }{\sqrt{{\left|\hat{\Phi }\left(\mathrm{0,0}\right)\right|}^2}}$，因为 $\epsilon$ 是任意小的，所以 $f=0$，则说明 $Clos{e}_{L^2\left(R^2\right)}{\displaystyle {\cup }_{j\in Z}}{V}_j=L^2\left(R^2\right)$。

4. 双正交二维四向加细函数和小波

定理5：如果二维四向加细函数是正交的，那么应该满足下列等式

$\begin{array}{l}\langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}\Phi \left(x-k\mathrm{,}y-l\right)\rangle ={\delta }_{\mathrm{0,}k}{\delta }_{\mathrm{0,}l}{I}_r\mathrm{,}\langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}\Phi \left(k-x\mathrm{,}y-l\right)\rangle =O_r\mathrm{,}\\ \langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}\Phi \left(x-k\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}\Phi \left(k-x\mathrm{,}l-y\right)\rangle =O_r\mathrm{.}\end{array}$

定理6：如果二维四向加细函数 $\Phi \left(x\mathrm{,}y\right)$ 和 $\tilde{\Phi }\left(x\mathrm{,}y\right)$ 是双正交的，那么应该满足下列等式

$\begin{array}{l}\langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}\tilde{\Phi }\left(x-k\mathrm{,}y-l\right)\rangle ={\delta }_{\mathrm{0,}k}{\delta }_{\mathrm{0,}l}{I}_r\mathrm{,}\langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}\tilde{\Phi }\left(k-x\mathrm{,}y-l\right)\rangle =O_r\mathrm{,}\\ \langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}\tilde{\Phi }\left(x-k\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}\tilde{\Phi }\left(k-x\mathrm{,}l-y\right)\rangle =O_r\mathrm{.}\end{array}$

定理7：如果二维四向加细函数 $\Phi \left(x\mathrm{,}y\right)$ 和 $\tilde{\Phi }\left(x\mathrm{,}y\right)$ 是双正交的尺度函数，那么它的双正面具，正负面具和双负面具符号都应该满足

$(\begin{array}{l}{\displaystyle \underset{k\mathrm{,}l=0}{\overset{a-1}{\sum }}}\{P^{+\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{+\mathrm{,}+}\left(-p_1\mathrm{,}-p_2\right)+P^{-\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{-\mathrm{,}+}\left(-p_1\mathrm{,}-p_2\right)\hfill \\ +P^{+\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{+\mathrm{,}-}\left(-p_1\mathrm{,}-p_2\right)+P^{-\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{-\mathrm{,}-}\left(-p_1\mathrm{,}-p_2\right)\}=I_r\hfill \\ {\displaystyle \underset{k\mathrm{,}l=0}{\overset{a-1}{\sum }}}\{P^{+\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{+\mathrm{,}-}\left(-p_1\mathrm{,}{p}_2\right)+P^{+\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{+\mathrm{,}+}\left(-p_1\mathrm{,}{p}_2\right)\hfill \\ +P^{-\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{-\mathrm{,}-}\left(-p_1\mathrm{,}{p}_2\right)+P^{-\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{-\mathrm{,}+}\left(-p_1\mathrm{,}{p}_2\right)\}=O_r\hfill \\ {\displaystyle \underset{k\mathrm{,}l=0}{\overset{a-1}{\sum }}}\{P^{+\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{-\mathrm{,}+}\left(p_1\mathrm{,}-p_2\right)+P^{+\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{-\mathrm{,}-}\left(p_1\mathrm{,}-p_2\right)\hfill \\ +P^{-\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{+\mathrm{,}+}\left(p_1\mathrm{,}-p_2\right)+P^{-\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{+\mathrm{,}-}\left(p_1\mathrm{,}-p_2\right)\}=O_r\hfill \\ {\displaystyle \underset{k\mathrm{,}l=0}{\overset{a-1}{\sum }}}\{P^{+\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{-\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right)+P^{+\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{-\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right)\hfill \\ +P^{-\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{+\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right)+P^{-\mathrm{,}-}\left(p_1\mathrm{,}{p}_2\right){\tilde{P}}^{+\mathrm{,}+}\left(p_1\mathrm{,}{p}_2\right)\}=O_r\hfill \end{array}$ (15)

其中： $p_1={\omega }_1+\frac{2k\pi }{a}$ ； $p_2={\omega }_2+\frac{2l\pi }{a}$。

证明：由文献 [7] 的定理1以及本文定理5定理6正交双正交定义易证。

假设 $\Phi \left(x\mathrm{,}y\right)$ 和 $\tilde{\Phi }\left(x\mathrm{,}y\right)$ 是双正交二维四向加细函数，对任意的 $j\in Z$，定义 $V_{j+1}=V_j\oplus W_j$，其中 $W_j$ 是 $V_j$ 在 $V_{j+1}$ 中的正交补。那么当 $j\ne k$ 时，就有 $W_j\perp W_k$ 并且 $L^2\mathrm{<mo\; stretchy="false">\; (\; </mo>}R\mathrm{<mo\; stretchy="false">\; )\; </mo>\; <mo>\; =\; </mo>}{{\displaystyle \oplus }}_{j\in Z}{W}_j$，其中 $W_j=\underset{h=1}{\overset{a-1}{{\displaystyle \oplus }}}\left\{W_j^{h\mathrm{,1}}\oplus W_j^{h\mathrm{,2}}\oplus W_j^{h\mathrm{,3}}\right\}$， $W_j^{h\mathrm{,1}}\perp W_j^{h\mathrm{,2}}$， $W_j^{h\mathrm{,1}}\perp W_j^{h\mathrm{,1}}$， $W_j^{h\mathrm{,2}}\perp W_j^{h\mathrm{,3}}$。

如果有r个小波函数 ${\psi }_1\left(x\right)\mathrm{,}{\psi }_2\left(x\right)\mathrm{,}\cdots \mathrm{,}{\psi }_r\left(x\right)$，则记 ${\Psi }^{h\mathrm{,}i}\left(x\right)={\left[{\psi }_1\left(x\right)\mathrm{,}{\psi }_2\left(x\right)\mathrm{,}\cdots \mathrm{,}{\psi }_r\left(x\right)\right]}^{\text{T}}$，则通过张量积的构造就有

${\Psi }^{h\mathrm{,}i}\left(x\mathrm{,}y\right)={\Psi }^{h\mathrm{,}i}\left(x\right)\psi \left(y\right)={\left[{\Psi }_1\left(x\mathrm{,}y\right)\mathrm{,}{\Psi }_2\left(x\mathrm{,}y\right)\mathrm{,}\cdots \mathrm{,}{\Psi }_r\left(x\mathrm{,}y\right)\right]}^{\text{T}}$

和

${\tilde{\Psi }}^{h\mathrm{,}i}\left(x\mathrm{,}y\right)={\left[{\tilde{\Psi }}_1\left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}_2\left(x\mathrm{,}y\right)\mathrm{,}\cdots \mathrm{,}{\tilde{\Psi }}_r\left(x\mathrm{,}y\right)\right]}^{\text{T}}$

$\left(h=1,2,\cdots ,a-1;i=1,2,3\right)$，使得集合 $\left\{{\Psi }^{h\mathrm{,}i}\left(x-k\mathrm{,}y-l\right)\mathrm{,}{\Psi }^{h\mathrm{,}i}\left(x-k\mathrm{,}l-y\right)\mathrm{,}{\Psi }^{h\mathrm{,}i}\left(k-x\mathrm{,}y-l\right)\mathrm{,}{\Psi }^{h\mathrm{,}i}\left(k-x\mathrm{,}l-y\right)\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\mathrm{,}h=\mathrm{1,2,}\cdots \mathrm{,}a-\mathrm{1\; <mo>\; ;\; </mo>}i=\mathrm{1,2,3}\right\}$ 和集合 $\left\{{\tilde{\Psi }}^{h\mathrm{,}i}\left(x-k\mathrm{,}y-l\right)\mathrm{,}{\tilde{\Psi }}^{h\mathrm{,}i}\left(x-k\mathrm{,}l-y\right)\mathrm{,}{\tilde{\Psi }}^{h\mathrm{,}i}\left(k-x\mathrm{,}y-l\right)\mathrm{,}{\tilde{\Psi }}^{h\mathrm{,}i}\left(k-x\mathrm{,}l-y\right)\mathrm{<mo>\; :\; </mo>}k\mathrm{,}l\in Z\mathrm{,}h=\mathrm{1,2,}\cdots \mathrm{,}a-\mathrm{1\; <mo>\; ;\; </mo>}i=\mathrm{1,2,3}\right\}$ 构成 $W_0$ 的一组双正交基，则 ${\Psi }^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 和 ${\tilde{\Psi }}^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 是与 $\Phi \left(x\mathrm{,}y\right)$ 和 $\tilde{\Phi }\left(x\mathrm{,}y\right)$ 对应的双正交二维四向多小波函数，应该满足

$\begin{array}{l}\tilde{\Phi }\left(x\mathrm{,}y\right)\mathrm{,}{\Psi }^{h\mathrm{,}i}\left(x-k\mathrm{,}y-l\right)=\langle \tilde{\Phi }\left(x\mathrm{,}y\right)\mathrm{,}{\Psi }^{h\mathrm{,}i}\left(x-k\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\\ \Phi \left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}^{h\mathrm{,}i}\left(k-x\mathrm{,}y-l\right)=\langle \Phi \left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}^{h\mathrm{,}i}\left(k-x\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\\ \tilde{\Phi }\left(x\mathrm{,}y\right)\mathrm{,}{\Psi }^{h\mathrm{,}i}\left(k-x\mathrm{,}y-l\right)=\langle \tilde{\Phi }\left(x\mathrm{,}y\right)\mathrm{,}{\Psi }^{h\mathrm{,}i}\left(k-x\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\\ \langle {\Psi }^{n\mathrm{,}m}\left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}^{s\mathrm{,}t}\left(x-k\mathrm{,}y-l\right)\rangle ={\delta }_{n\mathrm{,}s}{\delta }_{m\mathrm{,}t}{\delta }_{\mathrm{0,}k}{\delta }_{\mathrm{0,}l}{I}_r\mathrm{,}\end{array}$

$\begin{array}{l}\langle {\Psi }^{n\mathrm{,}m}\left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}^{s\mathrm{,}t}\left(x-k\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\\ \langle {\tilde{\Psi }}^{n\mathrm{,}m}\left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}^{s\mathrm{,}t}\left(x-k\mathrm{,}y-l\right)\rangle ={\delta }_{n\mathrm{,}s}{\delta }_{m\mathrm{,}t}{\delta }_{\mathrm{0,}k}{\delta }_{\mathrm{0,}l}{I}_r\mathrm{,}\\ \langle {\tilde{\Psi }}^{n\mathrm{,}m}\left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}^{s\mathrm{,}t}\left(x-k\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\\ \langle {\Psi }^{n\mathrm{,}m}\left(x\mathrm{,}y\right)\mathrm{,}{\Psi }^{s\mathrm{,}t}\left(k-x\mathrm{,}y-l\right)\rangle =O_r\mathrm{,}\end{array}$

$\begin{array}{l}\langle {\Psi }^{n\mathrm{,}m}\left(x\mathrm{,}y\right)\mathrm{,}{\Psi }^{s\mathrm{,}t}\left(k-x\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\\ \langle {\tilde{\Psi }}^{n\mathrm{,}m}\left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}^{s\mathrm{,}t}\left(k-x\mathrm{,}y-l\right)\rangle =O_r\mathrm{,}\\ \langle {\tilde{\Psi }}^{n\mathrm{,}m}\left(x\mathrm{,}y\right)\mathrm{,}{\tilde{\Psi }}^{s\mathrm{,}t}\left(k-x\mathrm{,}l-y\right)\rangle =O_r\mathrm{,}\end{array}$ (16)

其中： $h,n,s=1,2,\cdots ,a-1;i,m,t=1,2,3$。

假如 ${\Psi }^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 是 $\Phi \left(x\mathrm{,}y\right)$ 对应的多小波函数，相应的 ${\tilde{\Psi }}^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 是 $\tilde{\Phi }\left(x\mathrm{,}y\right)$ 对应的多小波函数，那么就存在 ${\left\{Q_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\right\}}_{k\mathrm{,}l\in Z}$， ${\left\{Q_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\right\}}_{k\mathrm{,}l\in Z}$， ${\left\{Q_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\right\}}_{k\mathrm{,}l\in Z}$， ${\left\{Q_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\right\}}_{k\mathrm{,}l\in Z}$ 和 ${\left\{{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\right\}}_{k\mathrm{,}l\in Z}$， ${\left\{{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\right\}}_{k\mathrm{,}l\in Z}$， ${\left\{{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}r\mathrm{,}-\mathrm{,}+}\right\}}_{k\mathrm{,}l\in Z}$， ${\left\{{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\right\}}_{k\mathrm{,}l\in Z}$，满足

$\begin{array}{c}{\Psi }^{h\mathrm{,}i}\left(x\mathrm{,}y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{Q}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\Phi \left(ax-k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{Q}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\Phi \left(ax-k\mathrm{,}l-ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{Q}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\Phi \left(k-ax\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{Q}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\Phi \left(k-ax\mathrm{,}l-ay\right)\mathrm{<mo>\; ;\; </mo>}\end{array}$ (17)

$\begin{array}{c}{\tilde{\Psi }}^{h\mathrm{,}i}\left(x\mathrm{,}y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\tilde{\Phi }\left(ax-k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\tilde{\Phi }\left(ax-k\mathrm{,}l-ay\right)\\ +{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\tilde{\Phi }\left(k-ax\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\tilde{\Phi }\left(k-ax\mathrm{,}l-ay\right)\mathrm{.}\end{array}$ (18)

对(17)式和(18)式两边做Fourier变换

$\begin{array}{c}{\hat{\Psi }}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)=Q^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)+Q^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\\ +Q^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)+Q^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\mathrm{<mo>\; ;\; </mo>}\end{array}$

$\begin{array}{c}{\widehat{\stackrel{˜}{\Psi }}}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{Q}}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)+{\tilde{Q}}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\\ +{\tilde{Q}}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)+{\tilde{Q}}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)\hat{\Phi }\left(-\frac{{\omega }_1}{a}\mathrm{,}-\frac{{\omega }_2}{a}\right)\mathrm{<mo>\; ;\; </mo>}\end{array}$

其中

$\begin{array}{l}{Q}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{Q}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}{Q}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{Q}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}\\ Q^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{Q}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}{Q}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{Q}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}\end{array}$

$\left(h=1,2,\cdots ,a-1;i=1,2,3\right)$ 为 ${\Psi }^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 的双正，正负，双负面具符号。

$\begin{array}{l}{\tilde{Q}}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}{\tilde{Q}}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}\\ {\tilde{Q}}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}{\tilde{Q}}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\left(\frac{{\omega }_1}{a}\mathrm{,}\frac{{\omega }_2}{a}\right)=\frac{1}{a^2}{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{Q}}_{k\mathrm{,}l}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}{\text{e}}^{-i\left(\frac{{\omega }_{1}k}{a}+\frac{{\omega }_{2}l}{a}\right)}\mathrm{,}\end{array}$

$\left(h=1,2,\cdots ,a-1;i=1,2,3\right)$ 为 ${\tilde{\Psi }}^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 的双正，正负，双负面具符号。

定理8：若 $\Phi \left(x\mathrm{,}y\right)$ 和 $\tilde{\Phi }\left(x\mathrm{,}y\right)$ 是双正交二维四向多加细函数， ${\Psi }^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 和 ${\tilde{\Psi }}^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 是相应的双正交二维四向多小波函数，则它们的面具符号满足

(19)

(20)

其中： $h,m,n=1,2,\cdots ,a-1;i,s,t=1,2,3;p_1={\omega }_1+\frac{2k\pi }{a};p_2={\omega }_2+\frac{2l\pi }{a}$。

证明：根据(16)式的正交性易得。

5. 构造算法

定理9：如果 $\Phi \left(x\mathrm{,}y\right)$ 和 $\tilde{\Phi }\left(x\mathrm{,}y\right)$ 是双正交二维四向多加细函数， ${\Psi }^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 和 ${\tilde{\Psi }}^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 是相应的双正交二维四向多小波函数， $P\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\tilde{P}\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 和 $Q^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{\tilde{Q}}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 是矩阵符号，构造

$(\begin{array}{l}{P}^{+\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\lambda }_0\left({\omega }_1\mathrm{,}{\omega }_2\right)P\left({\omega }_1\mathrm{,}{\omega }_2\right)+{\displaystyle \underset{j=1}{\overset{a-1}{\sum }}}{\lambda }_j\left({\omega }_1\mathrm{,}{\omega }_2\right)Q^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\hfill \\ P^{+\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\lambda }_a\left({\omega }_1\mathrm{,}{\omega }_2\right)P\left({\omega }_1\mathrm{,}-{\omega }_2\right)+{\displaystyle \underset{j=a+1}{\overset{2a-1}{\sum }}}{\lambda }_j\left({\omega }_1\mathrm{,}{\omega }_2\right)Q^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}-{\omega }_2\right)\mathrm{,}\hfill \\ P^{-\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\lambda }_{2a}\left({\omega }_1\mathrm{,}{\omega }_2\right)P\left(-{\omega }_1\mathrm{,}{\omega }_2\right)+{\displaystyle \underset{j=2a+1}{\overset{3a-1}{\sum }}}{\lambda }_j\left({\omega }_1\mathrm{,}{\omega }_2\right)Q^{h\mathrm{,}i}\left(-{\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\hfill \\ P^{-\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\lambda }_{3a}\left({\omega }_1\mathrm{,}{\omega }_2\right)P\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)+{\displaystyle \underset{j=3a+1}{\overset{4a-1}{\sum }}}{\lambda }_j\left({\omega }_1\mathrm{,}{\omega }_2\right)Q^{h\mathrm{,}i}\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)\mathrm{<mo>\; ;\; </mo>}\hfill \end{array}$

$(\begin{array}{l}{\tilde{P}}^{+\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{\lambda }}_0\left({\omega }_1\mathrm{,}{\omega }_2\right)\tilde{P}\left({\omega }_1\mathrm{,}{\omega }_2\right)+{\displaystyle \underset{j=1}{\overset{a-1}{\sum }}}{\tilde{\lambda }}_j\left({\omega }_1\mathrm{,}{\omega }_2\right){\tilde{Q}}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\hfill \\ {\tilde{P}}^{+\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{\lambda }}_a\left({\omega }_1\mathrm{,}{\omega }_2\right)\tilde{P}\left({\omega }_1\mathrm{,}-{\omega }_2\right)+{\displaystyle \underset{j=a+1}{\overset{2a-1}{\sum }}}{\tilde{\lambda }}_j\left({\omega }_1\mathrm{,}{\omega }_2\right){\tilde{Q}}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}-{\omega }_2\right)\mathrm{,}\hfill \\ {\tilde{P}}^{-\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{\lambda }}_{2a}\left({\omega }_1\mathrm{,}{\omega }_2\right)\tilde{P}\left(-{\omega }_1\mathrm{,}{\omega }_2\right)+{\displaystyle \underset{j=2a+1}{\overset{3a-1}{\sum }}}{\tilde{\lambda }}_j\left({\omega }_1\mathrm{,}{\omega }_2\right){\tilde{Q}}^{h\mathrm{,}i}\left(-{\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\hfill \\ {\tilde{P}}^{-\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{\lambda }}_{3a}\left({\omega }_1\mathrm{,}{\omega }_2\right)\tilde{P}\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)+{\displaystyle \underset{j=3a+1}{\overset{4a-1}{\sum }}}{\tilde{\lambda }}_j\left({\omega }_1\mathrm{,}{\omega }_2\right){\tilde{Q}}^{h\mathrm{,}i}\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)\mathrm{.}\hfill \end{array}$

$\left(h\mathrm{<mo>\; :\; </mo>0}\le h\le a-10\le h\le 4a-\mathrm{1\; <mo>\; ;\; </mo>}i\mathrm{<mo>\; :\; </mo>1}\le i\le \mathrm{3\; <mo>\; ;\; </mo>}i\mathrm{,}h\in Z^{+}\right)$。函数 ${\lambda }_j\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 和 ${\tilde{\lambda }}_j\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 以 $\frac{2\pi }{a}$ 为周期，且满足

1) $(\begin{array}{l}{\displaystyle \underset{j=0}{\overset{4a-1}{\sum }}}{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_j\left(-{\omega }_1,-{\omega }_2\right)=1,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{2a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{2a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{2a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\lambda }_0\left(0,0\right)+{\lambda }_a\left(0,0\right)+{\lambda }_{2a}\left(0,0\right)+{\lambda }_{3a}\left(0,0\right)=1,\hfill \\ {\tilde{\lambda }}_0\left(0,0\right)+{\tilde{\lambda }}_a\left(0,0\right)+{\tilde{\lambda }}_{2a}\left(0,0\right)+{\tilde{\lambda }}_{3a}\left(0,0\right)=1,\hfill \\ \left|{\lambda }_0\left(0,0\right)-{\lambda }_a\left(0,0\right)-{\lambda }_{2a}\left(0,0\right)+{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)-{\tilde{\lambda }}_a\left(0,0\right)-{\tilde{\lambda }}_{2a}\left(0,0\right)+{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\lambda }_0\left(0,0\right)+{\lambda }_a\left(0,0\right)-{\lambda }_{2a}\left(0,0\right)-{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)+{\tilde{\lambda }}_a\left(0,0\right)-{\tilde{\lambda }}_{2a}\left(0,0\right)-{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\lambda }_0\left(0,0\right)-{\lambda }_a\left(0,0\right)+{\lambda }_{2a}\left(0,0\right)-{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)-{\tilde{\lambda }}_a\left(0,0\right)+{\tilde{\lambda }}_{2a}\left(0,0\right)-{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1;\hfill \end{array}$

2) $(\begin{array}{l}{\displaystyle \underset{j=0}{\overset{4a-1}{\sum }}}{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_j\left(-{\omega }_1,-{\omega }_2\right)=1,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{2a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{2a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{2a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ \left|{\lambda }_0\left(0,0\right)+{\lambda }_a\left(0,0\right)+{\lambda }_{2a}\left(0,0\right)+{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)+{\tilde{\lambda }}_a\left(0,0\right)+{\tilde{\lambda }}_{2a}\left(0,0\right)+{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ {\lambda }_0\left(0,0\right)-{\lambda }_a\left(0,0\right)-{\lambda }_{2a}\left(0,0\right)+{\lambda }_{3a}\left(0,0\right)=1,\hfill \\ {\tilde{\lambda }}_0\left(0,0\right)-{\tilde{\lambda }}_a\left(0,0\right)-{\tilde{\lambda }}_{2a}\left(0,0\right)+{\tilde{\lambda }}_{3a}\left(0,0\right)=1,\hfill \\ \left|{\lambda }_0\left(0,0\right)+{\lambda }_a\left(0,0\right)-{\lambda }_{2a}\left(0,0\right)-{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)+{\tilde{\lambda }}_a\left(0,0\right)-{\tilde{\lambda }}_{2a}\left(0,0\right)-{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\lambda }_0\left(0,0\right)-{\lambda }_a\left(0,0\right)+{\lambda }_{2a}\left(0,0\right)-{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)-{\tilde{\lambda }}_a\left(0,0\right)+{\tilde{\lambda }}_{2a}\left(0,0\right)-{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1;\hfill \end{array}$

3) $(\begin{array}{l}{\displaystyle \underset{j=0}{\overset{4a-1}{\sum }}}{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_j\left(-{\omega }_1,-{\omega }_2\right)=1,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{2a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{2a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{2a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ \left|{\lambda }_0\left(0,0\right)+{\lambda }_a\left(0,0\right)+{\lambda }_{2a}\left(0,0\right)+{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)+{\tilde{\lambda }}_a\left(0,0\right)+{\tilde{\lambda }}_{2a}\left(0,0\right)+{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\lambda }_0\left(0,0\right)-{\lambda }_a\left(0,0\right)-{\lambda }_{2a}\left(0,0\right)+{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)-{\tilde{\lambda }}_a\left(0,0\right)-{\tilde{\lambda }}_{2a}\left(0,0\right)+{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ {\lambda }_0\left(0,0\right)+{\lambda }_a\left(0,0\right)-{\lambda }_{2a}\left(0,0\right)-{\lambda }_{3a}\left(0,0\right)=1,\hfill \\ {\tilde{\lambda }}_0\left(0,0\right)+{\tilde{\lambda }}_a\left(0,0\right)-{\tilde{\lambda }}_{2a}\left(0,0\right)-{\tilde{\lambda }}_{3a}\left(0,0\right)=1,\hfill \\ \left|{\lambda }_0\left(0,0\right)-{\lambda }_a\left(0,0\right)+{\lambda }_{2a}\left(0,0\right)-{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)-{\tilde{\lambda }}_a\left(0,0\right)+{\tilde{\lambda }}_{2a}\left(0,0\right)-{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1;\hfill \end{array}$

4) $(\begin{array}{l}{\displaystyle \underset{j=0}{\overset{4a-1}{\sum }}}{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_j\left(-{\omega }_1,-{\omega }_2\right)=1,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{2a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{2a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\lambda }_j\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{3a+j}\left(-{\omega }_1,-{\omega }_2\right)+{\lambda }_{a+j}\left({\omega }_1,{\omega }_2\right){\tilde{\lambda }}_{2a+j}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ \left|{\lambda }_0\left(0,0\right)+{\lambda }_a\left(0,0\right)+{\lambda }_{2a}\left(0,0\right)+{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)+{\tilde{\lambda }}_a\left(0,0\right)+{\tilde{\lambda }}_{2a}\left(0,0\right)+{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\lambda }_0\left(0,0\right)-{\lambda }_a\left(0,0\right)-{\lambda }_{2a}\left(0,0\right)+{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)-{\tilde{\lambda }}_a\left(0,0\right)-{\tilde{\lambda }}_{2a}\left(0,0\right)+{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\lambda }_0\left(0,0\right)+{\lambda }_a\left(0,0\right)-{\lambda }_{2a}\left(0,0\right)-{\lambda }_{3a}\left(0,0\right)\right|<1,\hfill \\ \left|{\tilde{\lambda }}_0\left(0,0\right)+{\tilde{\lambda }}_a\left(0,0\right)-{\tilde{\lambda }}_{2a}\left(0,0\right)-{\tilde{\lambda }}_{3a}\left(0,0\right)\right|<1,\hfill \\ {\lambda }_0\left(0,0\right)-{\lambda }_a\left(0,0\right)+{\lambda }_{2a}\left(0,0\right)-{\lambda }_{3a}\left(0,0\right)=1,\hfill \\ {\tilde{\lambda }}_0\left(0,0\right)-{\tilde{\lambda }}_a\left(0,0\right)+{\tilde{\lambda }}_{2a}\left(0,0\right)-{\tilde{\lambda }}_{3a}\left(0,0\right)=1;\hfill \end{array}$

则 $P^{+\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{P}^{+\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{P}^{-\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{P}^{-\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 和 ${\tilde{P}}^{+\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{\tilde{P}}^{+\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{\tilde{P}}^{-\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{\tilde{P}}^{-\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)$，产生一个双正交二维四向多细分函数 $\Phi \left(x\mathrm{,}y\right)$ 并满足

$\Phi \left(x\mathrm{,}y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}+}\Phi \left(ax-k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{+\mathrm{,}-}\Phi \left(ax-k\mathrm{,}l-ay\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}+}\Phi \left(k-ax\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{P}_{l\mathrm{,}k}^{-\mathrm{,}-}\Phi \left(k-ax\mathrm{,}l-ay\right)\mathrm{.}$

$\tilde{\Phi }\left(x\mathrm{,}y\right)={\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{P}}_{l\mathrm{,}k}^{+\mathrm{,}+}\tilde{\Phi }\left(ax-k\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{P}}_{l\mathrm{,}k}^{+\mathrm{,}-}\tilde{\Phi }\left(ax-k\mathrm{,}l-ay\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{P}}_{l\mathrm{,}k}^{-\mathrm{,}+}\tilde{\Phi }\left(k-ax\mathrm{,}ay-l\right)+{\displaystyle \underset{k\mathrm{,}l}{\sum }}{\tilde{P}}_{l\mathrm{,}k}^{-\mathrm{,}-}\tilde{\Phi }\left(k-ax\mathrm{,}l-ay\right)\mathrm{.}$

证明：将上面已经构造的 $P^{+\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{P}^{+\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{P}^{-\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{P}^{-\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 和 ${\tilde{P}}^{+\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{\tilde{P}}^{+\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{\tilde{P}}^{-\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{\tilde{P}}^{-\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)$，代入(15)式化简可得。再令(11)式 $P\left({\omega }_1\mathrm{,}{\omega }_2\right)=P\left(\mathrm{0,0}\right)$，则化简 $P\left(\mathrm{0,0}\right)$ 并求特征值可证得定理。

定理10：如果 $\Phi \left(x\mathrm{,}y\right)$ 和 $\tilde{\Phi }\left(x\mathrm{,}y\right)$ 是双正交二维四向多加细函数， ${\Psi }^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 和 ${\tilde{\Psi }}^{h\mathrm{,}i}\left(x\mathrm{,}y\right)$ 是相应的双正交二维四向多小波函数， $P\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\tilde{P}\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 和 $Q^{j\mathrm{,}s}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}{\tilde{Q}}^{j\mathrm{,}s}\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 是面具符号，构造

$(\begin{array}{l}{Q}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\mu }_0^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)P\left({\omega }_1\mathrm{,}{\omega }_2\right)+{\displaystyle \underset{j=1}{\overset{a-1}{\sum }}}{\mu }_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)Q^{j\mathrm{,}s}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\hfill \\ Q^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\mu }_a^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)P\left({\omega }_1\mathrm{,}-{\omega }_2\right)+{\displaystyle \underset{j=a+1}{\overset{2a-1}{\sum }}}{\mu }_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)Q^{j\mathrm{,}s}\left({\omega }_1\mathrm{,}-{\omega }_2\right)\mathrm{,}\hfill \\ Q^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\mu }_{2a}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)P\left(-{\omega }_1\mathrm{,}{\omega }_2\right)+{\displaystyle \underset{j=2a+1}{\overset{3a-1}{\sum }}}{\mu }_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)Q^{j\mathrm{,}s}\left(-{\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\hfill \\ Q^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\mu }_{3a}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)P\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)+{\displaystyle \underset{j=3a+1}{\overset{4a-1}{\sum }}}{\mu }_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)Q^{j\mathrm{,}s}\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)\mathrm{<mo>\; ;\; </mo>}\hfill \end{array}$

$(\begin{array}{l}{\tilde{Q}}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{\mu }}_0^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)\tilde{P}\left({\omega }_1\mathrm{,}{\omega }_2\right)+{\displaystyle \underset{j=1}{\overset{a-1}{\sum }}}{\tilde{\mu }}_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right){\tilde{Q}}^{j\mathrm{,}s}\left({\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\hfill \\ {\tilde{Q}}^{h\mathrm{,}i\mathrm{,}+\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{\mu }}_a^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)\tilde{P}\left({\omega }_1\mathrm{,}-{\omega }_2\right)+{\displaystyle \underset{j=a+1}{\overset{2a-1}{\sum }}}{\tilde{\mu }}_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right){\tilde{Q}}^{j\mathrm{,}s}\left({\omega }_1\mathrm{,}-{\omega }_2\right)\mathrm{,}\hfill \\ {\tilde{Q}}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}+}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{\mu }}_{2a}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)\tilde{P}\left(-{\omega }_1\mathrm{,}{\omega }_2\right)+{\displaystyle \underset{j=2a+1}{\overset{3a-1}{\sum }}}{\tilde{\mu }}_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right){\tilde{Q}}^{j\mathrm{,}s}\left(-{\omega }_1\mathrm{,}{\omega }_2\right)\mathrm{,}\hfill \\ {\tilde{Q}}^{h\mathrm{,}i\mathrm{,}-\mathrm{,}-}\left({\omega }_1\mathrm{,}{\omega }_2\right)={\tilde{\mu }}_{3a}^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)\tilde{P}\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)+{\displaystyle \underset{j=3a+1}{\overset{4a-1}{\sum }}}{\tilde{\mu }}_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right){\tilde{Q}}^{j\mathrm{,}s}\left(-{\omega }_1\mathrm{,}-{\omega }_2\right)\mathrm{.}\hfill \end{array}$

$\left(h\mathrm{<mo>\; :\; </mo>0}\le h\le a-10\le h\le 4a-\mathrm{1\; <mo>\; ;\; </mo>}i\mathrm{,}s\mathrm{<mo>\; :\; </mo>1}\le i\mathrm{,}s\le \mathrm{3\; <mo>\; ;\; </mo>}i\mathrm{,}h\mathrm{,}s\in Z^{+}\right)$。函数 ${\lambda }_j\left({\omega }_1\mathrm{,}{\omega }_2\right)$， ${\mu }_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 和 ${\tilde{\mu }}_j^{h\mathrm{,}i}\left({\omega }_1\mathrm{,}{\omega }_2\right)$ 以 $\frac{2\pi }{a}$ 为周期，且满足

1)

2) $(\begin{array}{l}{\displaystyle \underset{j=0}{\overset{4a-1}{\sum }}}{\mu }_j^{h,k}\left({\omega }_1,{\omega }_2\right){\tilde{\mu }}_j^{s,t}\left(-{\omega }_1,-{\omega }_2\right)={\delta }_{h,s}{\delta }_{k,t},\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\left\{{\mu }_j^{h,i}\left({\omega }_1,{\omega }_2\right){\tilde{\mu }}_{a+j}^{h,i}\left(-{\omega }_1,-{\omega }_2\right)+{\mu }_{2a+j}^{h,i}\left({\omega }_1,{\omega }_2\right){\tilde{\mu }}_{3a+j}^{h,i}\left(-{\omega }_1,-{\omega }_2\right)\right\}=0,\hfill \\ {\displaystyle \underset{j=0}{\overset{a-1}{\sum }}}\{{\mu }_j^{h,i}\hfill \end{array}$