Multi-Resolution Analysis Scaling Equation
Theorem
Let the function $\phi \in L^{2}(\mathbb{R})$ generate a multiresolution analysis. Then there exists a function $H_{0}\in L^{2}(0,1)$ of period $1$ satisfying the following equation.
$$ \begin{equation} \hat{\phi}(2\gamma) = H_{0}(\gamma)\hat{\phi}(\gamma),\quad \gamma \in \mathbb{R} \label{eq1} \end{equation} $$
This is called the scaling equation. Here $\hat{\phi}(\gamma)$ is the Fourier transform of $\phi$.
Explanation
Equation $(1)$ is also called the refinement equation. Moreover, a function $\phi$ satisfying the above theorem is called a scaling function, or one says that $\phi$ is refinable.
If the sequence of closed subspaces $\left\{V_{j}\right\}_{j \in \mathbb{Z}}$ of $L^{2}(\mathbb{R})$ and the function $\phi \in V_{0}$ satisfy the following conditions, then $\left( \left\{ V_{j} \right\}, \phi \right)$ is called a multiresolution analysis.
(a) For each $V_{j}$, $\cdots V_{-1} \subset V_{0} \subset V_{1}\cdots$ holds.
(b) $\overline{\cup_{j\in\mathbb{Z}}V_{j}}=L^{2}(\mathbb{R})$ and $\cap_{j\in\mathbb{Z}}V_{j}=\left\{ 0\right\}$.
(c) $\forall j\in \mathbb{Z}$, $V_{j+1}=D(V_{j})$.
(d) If $\forall k \in \mathbb{Z}$ and $f \in V_{0}$ then $T_{k}f \in V_{0}$.
(e) $\left\{ T_{k} \phi\right\}_{k\in \mathbb{Z}}$ is an orthonormal basis of $V_{0}$.
If $(\left\{ V_{j} \right\},\phi)$ is a multiresolution analysis, we say that $\phi$ generates the multiresolution analysis. $T_{k}$ is a translation, and $D$ is a dilation.
Proof
Since $\phi$ is assumed to generate a multiresolution analysis, by (e) $\left\{ T_{k}\phi \right\}_{k\in \mathbb{Z}}$ is an orthonormal basis of $V_{0}$. Considering the case $k=0$, we have $\phi \in V_{0}$. But by (a) $V_{0}\subset V_{1}$, hence $\phi \in V_{1}$. Also by (c) $V_{1}=D(V_{0})$, therefore
$$ D^{-1}\phi \in V_{0} $$
Since $V_{0}$ is a vector space, it is closed under scalar multiplication. Thus
$$ \frac{1}{\sqrt{2}}D^{-1}\phi \in V_{0} $$
Because $\left\{ T_{-k}\phi \right\}_{k\in \mathbb{Z}}$ was an orthonormal basis of $V_{0}$, there exist coefficients $\left\{ c_{k} \right\}$ so that we can express as follows.
$$ \begin{equation} \frac{1}{\sqrt{2}}D^{-1}\phi = \sum _{k\in \mathbb{Z}} c_{k}T_{-k}\phi \label{eq2} \end{equation} $$
Lemma
Let $T$ be a bounded linear operator on the normed space $V$. Let $\left\{ \mathbf{v}_{k} \right\}_{k=1}^{\infty}$ be a sequence of elements of $V$. If for some constant $\left\{ c_{k} \right\}_{k=1}^{\infty}$ the sequence $\sum_{k=1}^{\infty}c_{k}\mathbf{v}_{k}$ converges, then $$ T\sum_{k=1}^{\infty}c_{k}\mathbf{v}_{k}=\sum_{k=1}^{\infty}c_{k}T\mathbf{v}_{k} $$ holds.
Now apply the Fourier transform to both sides of $(2)$. Then, by the above lemma,
$$ \begin{align*} \frac{1}{\sqrt{2}}\mathcal{F}D^{-1}\phi&= \mathcal{F}\sum _{k\in \mathbb{Z}} c_{k}T _{-k}\phi \\ &= \sum _{k\in \mathbb{Z}} c_{k}\mathcal{F}T _{-k}\phi \end{align*} $$
Here $\mathcal{F}D^{-1}=D\mathcal{F}$ and $\mathcal{F}T_{-k}=E_{k}\mathcal{F}$, hence
$$ \frac{1}{\sqrt{2}}D \hat{\phi}(\gamma) = \sum _{k\in \mathbb{Z}}c_{k}E_{k}\hat{\phi}(\gamma) $$
Now applying dilation and modulation gives
$$ \hat{\phi}(2\gamma) =\sum _{k \in \mathbb{Z}}c_{k}e^{2\pi i k \gamma}\hat{\phi}(\gamma) $$
If we define it by $H_{0}(\gamma) := \sum \limits_{k \in \mathbb{Z}}c_{k}e^{2\pi i k \gamma}$, it becomes a function with period $1$. Therefore
$$ \hat{\phi}(2\gamma) = H_{0}(\gamma)\hat{\phi}(\gamma),\quad \gamma \in \mathbb{R} $$
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