引言

相比经典小波, 非平面域上小波和小波框架能有效处理和分析数量庞大的非结构化数据集，在实际应用中也取得了理想的结果^[1-2]。Hammond等^[3]利用图谱理论推广了经典小波概念，首次提出频谱图小波变换(SGWT)，开启了非平面域上小波研究的新阶段。Narang等^[4]利用图正交镜像滤波器消除了二部图中谱折叠产生的混叠现象，推导出二部图滤波器组正交性的充要条件，通过刻画二部图两通道小波滤波器组的完全重构，实现了对定义在任意有限无向赋权二部图顶点上的函数分析。Dong^[5]基于多分辨分析(MRA)引入了流形紧小波框架和图紧小波框架的刻画，相比频谱图小波变换，实例验证了快速图紧小波框架变换(WFTG)处理图数据的高效性。

但是以上文献主要研究非平面域上小波和紧小波框架，关于流形小波双框架的研究还未见报道。本文受文献[4]和[5]的研究启发，在文献[5]的工作基础上进一步研究，将其刻画思想推广到小波双框架，首先在流形上提出了小波双框架概念；其次，利用多分辨分析(MRA)滤波器组，重点研究了流形上小波双框架的构造和刻画，得到了一些等价性质。

1 流形小波双框架的相关概念

设流形$\mathscr{M}$是一个概率测度为μ(μ($\mathscr{M}$)=1), 光滑边界(或边界是空集)为S的d(d≥2)维紧致连续光滑黎曼流形。空间L₂($\mathscr{M}$):=L₂($\mathscr{M}$, μ)是流形$\mathscr{M}$上关于测度μ的复值平方可积函数空间，其中f∈L₂($\mathscr{M}$)的L₂范数为‖f‖_{L2($\mathscr{M}$)}=($\int_{\mathscr{M}}$|f(x)|²dμ(x))^1/2。对于f和g∈L₂($\mathscr{M}$)，L₂($\mathscr{M}$)是一个具有内积〈f, g〉_{L2($\mathscr{M}$)}=$\int_{\mathscr{M}}$f(x)g(x)dμ(x)的希尔伯特空间，其中g是函数g的共轭。

将序列对{(u_$\ell$, λ_$\ell$)}_$\ell$=0^∞称为空间L₂($\mathscr{M}$)的正交特征对，其中序列{u_$\ell$}_$\ell$=0^∞是u₀≡1的空间L₂($\mathscr{M}$)的正交基，序列{λ_$\ell$}_$\ell$=0^∞是满足0=λ₀≤λ₁≤…和$\mathop {\lim }\limits_{\ell \to \infty } {\mkern 1mu} $λ_$\ell$=∞的非负递增数列。一个典型特例就是，{(u_$\ell$, λ_$\ell$)}_$\ell$=0^∞是流形$\mathscr{M}$上满足关系式Δu_$\ell$=-λ_$\ell$²u_$\ell$的Laplace-Beltrami算子Δ的特征函数和特征值集合。函数f∈L₂($\mathscr{M}$)的傅里叶变换${{{\hat{f}}}_\ell}$为${{{\hat{f}}}_\ell}$:=〈f, u_$\ell$〉_{L2($\mathscr{M}$)}。设L₁($\mathbb{R}$)是实数集$\mathbb{R}$上关于勒贝格测度的绝对值可积函数空间，Ψ:={α; β¹, …, β^r}是空间L₁($\mathbb{R}$)中的函数族。

类似于经典小波集，连续框架小波φ_{j, y}(x)和ψ_{j, y}ⁿ(x)在尺度j处的伸缩和顶点y∈$\mathscr{M}$处的平移分别如式(1)、(2)^[6-7]

$ {\varphi _{j,\mathit{\boldsymbol{y}}}}\left( \mathit{\boldsymbol{x}} \right) = \sum\limits_{\ell = 0}^\infty {\hat \alpha \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\bar u}_\ell }\left( \mathit{\boldsymbol{y}} \right){u_\ell }\left( \mathit{\boldsymbol{x}} \right)} $

(1)

$ \psi _{j,\mathit{\boldsymbol{y}}}^n\left( \mathit{\boldsymbol{x}} \right) = \sum\limits_{\ell = 0}^\infty {{{\hat \beta }^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\bar u}_\ell }\left( \mathit{\boldsymbol{y}} \right){u_\ell }\left( \mathit{\boldsymbol{x}} \right)} ,n = 1, \cdots ,r $

(2)

流形$\mathscr{M}$上的连续框架小波系统CFS_J(Ψ)(始于尺度J∈$\mathbb{Z}$)是一个如式(3)的非齐次仿射系统^[8-9]

$ \begin{array}{l} CF{S_J}\left( \mathit{\Psi } \right) = CF{S_J}\left\{ {\alpha ;{\beta ^1}, \cdots ,{\beta ^r}} \right\} = \left\{ {{\varphi _{J,\mathit{\boldsymbol{y}}}}:\mathit{\boldsymbol{y}} \in \mathscr{M}} \right\} \cup \\ \left\{ {\psi _{j,\mathit{\boldsymbol{y}}}^1, \cdots ,\psi _{j,\mathit{\boldsymbol{y}}}^r:\mathit{\boldsymbol{y}} \in \mathscr{M},j \ge J} \right\} \end{array} $

(3)

定义1  设Ψ={α; β¹, …, β^r}⊂L₁($\mathbb{R}$)和${\mathit{\tilde \Psi }}=\left\{ \tilde{\alpha };{{{\tilde{\beta }}}^{1}}, \cdots, {{{\tilde{\beta }}}^{r}} \right\}\subset {{L}_{1}}\left( \mathbb{R} \right)$是两列小波父母向量函数，如果条件1)~3)同时满足

1) CFS_J(Ψ)和CFS_J(${\mathit{\tilde \Psi }}$)在空间L₂($\mathscr{M}$)中；

2) CFS_J(Ψ)和CFS_J(${\mathit{\tilde \Psi }}$)都是空间L₂($\mathscr{M}$)中的小波框架；

3) 在L₂-范数下，任意函数f∈L₂($\mathscr{M}$)都满足完全重构表达式

$ \begin{array}{l} f = \int_\mathscr{M} {\left\langle {f,{\varphi _{J,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{J,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} + \sum\limits_{j = J}^\infty {\sum\limits_{n = 1}^r {\int_\mathscr{M} {\left\langle {f,\psi _{j,\mathit{\boldsymbol{y}}}^n} \right\rangle } } } \\ \tilde \psi _{j,\mathit{\boldsymbol{y}}}^n{\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right) \end{array} $

(4)

或者等价于式(5)

$ \begin{array}{l} \left\| f \right\|_{{L_2}\left( \mathscr{M} \right)}^2 = \int_\mathscr{M} {\left\langle {f,{\varphi _{J,\mathit{\boldsymbol{y}}}}} \right\rangle \left\langle {{{\tilde \varphi }_{J,\mathit{\boldsymbol{y}}}},f} \right\rangle {\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} + \\ \sum\limits_{j = J}^\infty {\sum\limits_{n = 1}^r {\int_\mathscr{M} {\left\langle {f,\psi _{j,\mathit{\boldsymbol{y}}}^n} \right\rangle \left\langle {\tilde \psi _{j,\mathit{\boldsymbol{y}}}^n,f} \right\rangle {\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} } } \end{array} $

(5)

则称系统对(CFS_J(Ψ), CFS_J(${\mathit{\tilde \Psi }}$))是空间L₂($\mathscr{M}$)中的连续小波双框架。特别地，若Ψ=${\mathit{\tilde \Psi }}$，则称系统对(CFS_J(Ψ), CFS_J(${\mathit{\tilde \Psi }}$))是空间L₂($\mathscr{M}$)中的连续紧小波框架，记作CFS_J(Ψ)。

本文主要研究的是空间L₂($\mathscr{M}$)中基于MRA的小波双框架构造，所以在此引入MRA滤波器组η:={a; b₁, …, b_r}⊂l₁($\mathbb{Z}$)和$\tilde{\eta }:=\left\{ \tilde{a};{{{\tilde{b}}}_{1}}, \cdots, {{{\tilde{b}}}_{r}} \right\}\subset {{l}_{1}}\left( \mathbb{Z} \right)$，使得它们分别能完全确定Ψ和${\mathit{\tilde \Psi }}$，即(Ψ, ${\mathit{\tilde \Psi }}$)中函数傅里叶变换和(η, ${\tilde{\eta }}$)中滤波器傅里叶级数满足关系式(6)和(7)

$ \hat \alpha \left( {2\xi } \right) = \hat \alpha \left( \xi \right)\hat \alpha \left( \xi \right),{{\hat \beta }^n}\left( {2\xi } \right) = {{\hat b}_n}\left( \xi \right)\hat \alpha \left( \xi \right) $

(6)

$ \begin{align} & \ \ \ \ \hat{\tilde{\alpha }}\left( 2\xi \right)=\hat{\tilde{\alpha }}\left( \xi \right)\hat{\tilde{\alpha }}\left( \xi \right),{{{\hat{\tilde{\beta }}}}^{n}}\left( 2\xi \right)={{{\hat{\tilde{b}}}}_{n}}\left( \xi \right)\hat{\tilde{\alpha }}\left( \xi \right),n= \\ & 1,\cdots ,r,\xi \in \mathbb{R} \\ \end{align} $

(7)

2 流形小波双框架的等价刻画

基于文献[5]中流形紧小波框架的刻画研究，本文得出定理1。

定理1  设J₀∈ $\mathbb{Z}$是整数，η:={a; b₁, …, b_r}⊂ l₁($\mathbb{Z}$)和$\tilde{\eta }:=\left\{ \tilde{a};{{{\tilde{b}}}_{1}}, \cdots, {{{\tilde{b}}}_{r}} \right\}\subset {{l}_{1}}\left( \mathbb{Z} \right)$分别是系统Ψ:= {α; β¹, …, β^r}⊂L₁($\mathbb{R}$)和${\mathit{\tilde \Psi }}=\left\{ \tilde{\alpha };{{{\tilde{\beta }}}^{1}}, \cdots, {{{\tilde{\beta }}}^{r}} \right\}\subset {{L}_{1}}\left( \mathbb{R} \right)$的MRA结合系数，其中r≥1；CFS_J(Ψ)和CFS_J(${\mathit{\tilde \Psi }}$)是具有框架小波φ_{j, y}，${\tilde \varphi } _{j, \mathit{\boldsymbol{y}}}$，ψ_{j, y}ⁿ，${\mathit{\tilde \psi }}_{j, \mathit{\boldsymbol{y}}}^{n}$的连续框架小波系统，其中J≥J₀。假设${\hat{\alpha }}$和${\hat{\tilde{\alpha }}}$在原点处连续，对于任意的y∈$\mathscr{M}$，n=1, …, r和j≥J₀，都有：φ_{j, y}，${\tilde \varphi } _{j, \mathit{\boldsymbol{y}}}^{n}$，ψ_{j, y}ⁿ，${\mathit{\tilde \psi }}_{j, \mathit{\boldsymbol{y}}}$是空间L₂($\mathscr{M}$)中的函数，且相关系统CFS_J(Ψ)和CFS_J(${\mathit{\tilde \Psi }}$)也是空间L₂($\mathscr{M}$)中的小波框架。那么，下列结论(ⅰ)~(ⅴ)彼此等价。

(ⅰ)对于任意J≥J₀，系统对(CFS_J(Ψ), CFS_J(${\mathit{\tilde \Psi }}$))是空间L₂($\mathscr{M}$)中的连续小波双框架；

(ⅱ)对于任意f∈L₂($\mathscr{M}$)，恒等式(8)、(9)成立

$ \mathop {\lim }\limits_{j \to \infty } {\left\| {\int_\mathscr{M} {\left\langle {f,{\varphi _{J,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{J,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right) - f} } \right\|_{{L_2}\left( \mathscr{M} \right)}} = 0 $

(8)

$ \begin{array}{l} \int_\mathscr{M} {\left\langle {f,{\varphi _{j + 1,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{j + 1,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} = \int_\mathscr{M} {\left\langle {f,{\varphi _{J,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{J,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu } \\ \left( \mathit{\boldsymbol{y}} \right) + \int_\mathscr{M} {\sum\limits_{n = 1}^r {\left\langle {f,\psi _{j,\mathit{\boldsymbol{y}}}^n} \right\rangle \tilde \psi _{j,\mathit{\boldsymbol{y}}}^n{\rm{d}}\mu } \left( \mathit{\boldsymbol{y}} \right)} ,j \ge {J_0} \end{array} $

(9)

(ⅲ)对于任意f∈L₂($\mathscr{M}$)，恒等式(10)、(11)成立

$ \mathop {\lim }\limits_{j \to \infty } \int_\mathscr{M} {\left\langle {f,{\varphi _{j,\mathit{\boldsymbol{y}}}}} \right\rangle \left\langle {{{\tilde \varphi }_{j,\mathit{\boldsymbol{y}}}},f} \right\rangle {\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} = \left\| f \right\|_{{L_2}\left( \mathscr{M} \right)}^2 $

(10)

$ \begin{array}{l} \int_\mathscr{M} {\left\langle {f,{\varphi _{j + 1,\mathit{\boldsymbol{y}}}}} \right\rangle \left\langle {{{\tilde \varphi }_{j + 1,\mathit{\boldsymbol{y}}}},f} \right\rangle {\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} = \int_\mathscr{M} {\left\langle {f,{\varphi _{J,\mathit{\boldsymbol{y}}}}} \right\rangle \left\langle {{{\tilde \varphi }_{J,\mathit{\boldsymbol{y}}}},} \right.} \\ \left. f \right\rangle {\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right) + \int_\mathscr{M} {\sum\limits_{n = 1}^r {\left\langle {f,\psi _{j,\mathit{\boldsymbol{y}}}^n} \right\rangle \left\langle {\tilde \psi _{j,\mathit{\boldsymbol{y}}}^n,f} \right\rangle {\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} } ,j \ge {J_0} \end{array} $

(11)

(ⅳ)Ψ和${\mathit{\tilde \Psi }}$中的函数满足

$ \mathop {\lim }\limits_{j \to \infty } \bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right) = 1,\ell \ge 0 $

(12)

$ \begin{array}{l} \bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right) + \sum\limits_{n = 1}^r {{{\bar {\hat \beta} }^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\hat {\tilde \beta }}^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)} = \bar {\hat \alpha} \\ \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right),\ell \ge 0,j \ge {J_0} \end{array} $

(13)

(ⅴ)尺度函数α和${\tilde{\alpha }}$满足式(12)，且滤波器组η和${\tilde{\eta }}$中的滤波器满足式(14)

$ \begin{array}{l} \bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right) + \sum\limits_{n = 1}^r {{{\bar {\hat b}}_n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\bar {\hat b}}_n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)} = 1,\\ \forall \ell \in \sigma _{\bar {\hat \alpha} ,\hat {\tilde \alpha} }^j,\forall j \ge {J_0} + 1 \end{array} $

(14)

其中$\sigma _{\bar{\hat{\alpha }}, \hat{\tilde{\alpha }}}^{j}=\left\{ \ell \in {{\mathbb{N}}_{0}}:\bar{\hat{\alpha }}\left( \frac{{{\lambda }_{\ell }}}{{{2}^{j}}} \right)\hat{\tilde{\alpha }}\left( \frac{{{\lambda }_{\ell }}}{{{2}^{j}}} \right)\ne 0 \right\}, $${{\mathbb{N}}_{0}}=\mathbb{N}\cup \left\{ 0 \right\}$。

证明  情况① (ⅰ)$\Leftrightarrow $(ⅱ)。

对于任意J≥J₀，系统对(CFS_J(Ψ), CFS_J(${\mathit{\tilde \Psi }}$))都是空间L₂($\mathscr{M}$)中的连续小波双框架，则对任意f∈L₂($\mathscr{M}$)，J≥J₀，等式(15)都成立

$ \begin{array}{l} f = \int_\mathscr{M} {\left\langle {f,{\varphi _{J,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{J,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu } \left( \mathit{\boldsymbol{y}} \right) + \sum\limits_{j = J}^\infty {\sum\limits_{n = 1}^r {\int_\mathscr{M} {\left\langle {f,\psi _{j,\mathit{\boldsymbol{y}}}^n} \right\rangle } } } \\ \tilde \psi _{j,\mathit{\boldsymbol{y}}}^n{\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right) = \int_\mathscr{M} {\left\langle {f,{\varphi _{J + 1,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{J + 1,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu } \left( \mathit{\boldsymbol{y}} \right) + \sum\limits_{j = J + 1}^\infty {\sum\limits_{n = 1}^r {} } \\ \int_\mathscr{M} {\left\langle {f,\psi _{j,\mathit{\boldsymbol{y}}}^n} \right\rangle \tilde \psi _{j,\mathit{\boldsymbol{y}}}^n{\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} \end{array} $

(15)

同时可有

$ \begin{array}{l} \int_\mathscr{M} {\left\langle {f,{\varphi _{J + 1,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{J + 1,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu } \left( \mathit{\boldsymbol{y}} \right) = \int_\mathscr{M} {\left\langle {f,{\varphi _{J,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{J,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu } \\ \left( \mathit{\boldsymbol{y}} \right) + \sum\limits_{n = 1}^r {\int_\mathscr{M} {\left\langle {f,\psi _{J,\mathit{\boldsymbol{y}}}^n} \right\rangle \tilde \psi _{J,\mathit{\boldsymbol{y}}}^n{\rm{d}}\mu \left( \mathit{\boldsymbol{y}} \right)} } \end{array} $

故式(9)成立。

为方便计算，定义算子$\mathscr{P}{{\nu}_{j}}$和$\mathscr{P}w_{j}^{n}$如式(16)

$ \left\{ \begin{array}{l} \mathscr{P}{\nu _j}\left( f \right): = \int_\mathscr{M} {\left\langle {f,{\varphi _{j,\mathit{\boldsymbol{y}}}}} \right\rangle {{\tilde \varphi }_{j,\mathit{\boldsymbol{y}}}}{\rm{d}}\mu } \left( \mathit{\boldsymbol{y}} \right)\\ \mathscr{P}w_j^n\left( f \right): = \int_\mathscr{M} {\left\langle {f,\psi _{j,\mathit{\boldsymbol{y}}}^n} \right\rangle \tilde \psi _{j,\mathit{\boldsymbol{y}}}^n{\rm{d}}\mu } \left( \mathit{\boldsymbol{y}} \right) \end{array} \right.,f \in {L_2}\left( \mathscr{M} \right) $

(16)

由式(9)得

$ \mathscr{P}{\nu _{m + 1}}\left( f \right) = \mathscr{P}{\nu _J}\left( f \right) + \sum\limits_{j = J}^m {\sum\limits_{n = 1}^r {\mathscr{P}w_j^n\left( f \right)} } ,\forall m \ge J,\forall J \ge {J_0} $

(17)

令m→∞，则在L₂-范数下有

$ \mathop {\lim }\limits_{m \to \infty } \mathscr{P}{\nu _{m + 1}}\left( f \right) = \mathscr{P}{\nu _J}\left( f \right) + \sum\limits_{j = J}^\infty {\sum\limits_{n = 1}^r {\mathscr{P}w_j^n\left( f \right)} } = f $

故式(8)成立，进一步可得(ⅰ)$\Rightarrow $(ⅱ)。

相反地，由式(9)显然有式(17)成立，通过式(8)在式(17)等号两边对m取极限，有

$ \mathop {\lim }\limits_{m \to \infty } \mathscr{P}{\nu _{m + 1}}\left( f \right) = f = \mathscr{P}{\nu _J}\left( f \right) + \sum\limits_{j = J}^\infty {\sum\limits_{n = 1}^r {\mathscr{P}w_j^n\left( f \right)} } $

由此可得(ⅱ)$\Rightarrow $(ⅰ)成立。情况①得证。

情况② (ⅱ)$\Leftrightarrow $(ⅲ)。

(ⅱ)与(ⅲ)之间的等价性可由极化恒等式性质直接得到。情况②得证。

情况③ (ⅱ)$\Leftrightarrow $(ⅳ)。

通过式(1)、(2)和u_$\ell$的正交性，可以得到式(18)、(19)

$ \begin{array}{l} \left\langle {f,{\varphi _{j,\mathit{\boldsymbol{y}}}}} \right\rangle = \left\langle {f,\sum\limits_{\ell = 0}^\infty {\hat \alpha \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\bar u}_\ell }\left( \mathit{\boldsymbol{y}} \right){u_\ell }} } \right\rangle = \sum\limits_{\ell = 0}^\infty {\bar {\hat \alpha} } \\ \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){u_\ell }\left( \mathit{\boldsymbol{y}} \right){\hat f_\ell } \end{array} $

(18)

$ \begin{array}{l} \left\langle {f,\psi _{j,\mathit{\boldsymbol{y}}}^n} \right\rangle = \left\langle {f,\sum\limits_{\ell = 0}^\infty {{{\hat \beta }^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\bar u}_\ell }\left( \mathit{\boldsymbol{y}} \right){u_\ell }} } \right\rangle = \sum\limits_{\ell = 0}^\infty {} \\ {{\bar {\hat \beta }}^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){u_\ell }\left( \mathit{\boldsymbol{y}} \right){{\hat f}_\ell } \end{array} $

(19)

再结合式(1)、(2)、(18)和(19)，式(16)在傅里叶域中可以等价为式(20)

$ \left\{ \begin{gathered} {\left( {\mathscr{P}{{\hat \nu }_j}\left( f \right)} \right)_\ell } = \bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\hat f}_\ell } \hfill \\ {\left( {\mathscr{P}\hat w_j^n\left( f \right)} \right)_\ell } = {{\bar {\hat \beta }}^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\hat {\tilde \beta }}^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\hat f}_\ell } \hfill \\ \end{gathered} \right.,\ell \in {\mathbb{N}_0} $

(20)

故通过式(20)和帕塞瓦夫恒等式可以得到

$ \begin{array}{*{20}{c}} {\left\| {\mathscr{P}{{\hat \nu }_j}\left( f \right) - f} \right\|_{{L_2}\left( \mathscr{M} \right)}^2 = }\\ {\sum\limits_{\ell = 0}^\infty {{{\left( {\bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right) - 1} \right)}^2}{{\left| {{{\hat f}_\ell }} \right|}^2}} } \end{array} $

(21)

那么在条件(8)下，令式(21)中的j→∞，则有

$ \begin{array}{l} \mathop {\lim }\limits_{j \to \infty } \left\| {\mathscr{P}{\nu _j}\left( f \right) - f} \right\|_{{L_2}\left( \mathscr{M} \right)}^2 = \mathop {\lim }\limits_{j \to \infty } \sum\limits_{\ell = 0}^\infty {\left( {\bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)\hat {\tilde \alpha} } \right.} \\ {\left. {\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right) - 1} \right)^2}{\left| {{{\hat f}_\ell }} \right|^2} = 0 \end{array} $

所以当j→∞时，式(21)等号右边的每一项都必须趋向于0，即对于任意$\ell$≥0，有$\mathop {\lim }\limits_{j \to \infty } {\mkern 1mu} \bar{\hat{\alpha }}\left( \frac{{{\lambda }_{\ell }}}{{{2}^{j}}} \right)\hat{\tilde{\alpha }}\left( \frac{{{\lambda }_{\ell }}}{{{2}^{j}}} \right)=1$。所以式(8)$\Rightarrow $式(12)。

另一方面，当任意$\ell$≥0且$\mathop {\lim }\limits_{j \to \infty } {\mkern 1mu} \bar{\hat{\alpha }}\left( \frac{{{\lambda }_{\ell }}}{{{2}^{j}}} \right)\hat{\tilde{\alpha }}\left( \frac{{{\lambda }_{\ell }}}{{{2}^{j}}} \right)=1$时，由${\hat{\alpha }}$和${\hat{\tilde{\alpha }}}$在原点处的连续性和勒贝格控制收敛定理，有$\mathop {\lim }\limits_{j \to \infty } {\mkern 1mu} \left\| \mathscr{P}{{\mathit{\boldsymbol{v}}}_{j}}\left( f \right)-f \right\|_{{{L}_{2}}\left( \mathscr{M} \right)}^{2}$，故式(12)$\Rightarrow $式(8)。根据式(20)，式(9)在傅里叶域中可等价为

$ \begin{gathered} \bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right){{\hat f}_\ell } = \bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\hat f}_\ell } + \sum\limits_{n = 1}^r {{{\bar {\hat \beta} }^n}} \hfill \\ \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\hat {\tilde \beta} }^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\hat f}_\ell },\ell \in {\mathbb{N}_0} \hfill \\ \end{gathered} $

(22)

由式(22)得出(9)$\Rightarrow $(13)。

同时，因任意函数f∈L₂($\mathscr{M}$)在空间L₂($\mathscr{M}$)中都有傅里叶展开式$f=\sum\limits_{\ell =0}^{\infty }{{{{\hat{f}}}_{\ell }}{{u}_{\ell }}}$，则结合式(13)和(20)可推出式(9)成立。

由此可得(ⅱ)$\Leftrightarrow $(ⅳ)。情况③得证。

情况④ (ⅳ)$\Leftrightarrow $(ⅴ)。

由式(6)、(7)中的关系式可得，对于$\ell$≥0和j≥J₀，都有

$ \begin{array}{l} \bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right) + \sum\limits_{n = 1}^r {{{\bar {\hat \beta }}^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right){{\hat {\tilde \beta} }^n}\left( {\frac{{{\lambda _\ell }}}{{{2^j}}}} \right)} = \bar {\hat \alpha} \\ \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right) + \sum\limits_{n = 1}^r {{{\bar {\hat b}}_n}\left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\bar {\hat \alpha} } \\ \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right){{\hat {\tilde b}}_n}\left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right) = \left[ {\bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right) + } \right.\\ \left. {\sum\limits_{n = 1}^r {{{\bar {\hat b}}_n}\left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right){{\hat {\tilde b}}_n}\left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)} } \right]\bar {\hat \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right)\hat {\tilde \alpha} \left( {\frac{{{\lambda _\ell }}}{{{2^{j + 1}}}}} \right) \end{array} $

于是有式(13)等价于式(14)，从而(ⅳ)$\Leftrightarrow $(ⅴ)。情况④得证。

综合情况①~④的证明，可得定理1成立。

定理1表明，形如式(3)的连续框架系统对序列{(CFS_J(${\mathit{\tilde \Psi }}$), CFS_J(Ψ))}_J=0^∞是L₂($\mathscr{M}$)中的小波双框架序列的完全等价刻画。

值得注意的是，由定理1中结论(ⅳ)~(ⅴ)可知，根据酉扩展原理构造的所有L₂($\mathbb{R}$)中的小波双框架同样可生成空间L₂($\mathscr{M}$)中的小波双框架，这极大地简化了流形上小波双框架的构造。

3 结束语

本文受经典双框架理论启发提出了流形小波双框架的概念，作为流形紧框架的推广，在相关小波双框架滤波器组基础上发展了流形小波双框架理论。定理1提供了空间L₂($\mathscr{M}$)中一列框架系统对是小波双框架序列的完全刻画，为后续非平面域信号处理奠定了理论基础。