where can I find the proof of this theorem of wavelet frames?

Question

Let $\mathbb{K}$ be a local field.

Definition of local field:

Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ and $\mathbb{K}^*$ are locally compact Abelian groups, where $\mathbb{K}^+$ and $\mathbb{K}^*$ denote the additive and multiplicative groups of $\mathbb{K}$, respectively.

a local field is a locally compact field with respect to a non-discrete topology.

For given $\Psi:=\left \{ \psi_{1},...,\psi_{L} \right \}\subset L^{2}(\mathbb{K})$, define the wavelet system

$X(\Psi ):=\left \{ \psi_{l,j,k}:1\leq l\leq L;j\in \mathbb{Z},k\in \mathbb{N}_{0} \right \}$ where $\psi_{j,k}^{l}=q^{\frac{j}{2}}\psi^{l}(\rho ^{j}.-u(k))$. ($q$ and $\rho$ is fixed.)

the wavelet system $X(\Psi )$ is called a Parseval wavelet frame if $\sum_{l=1}^{L}\sum_{j\in \mathbb{Z}}\sum_{k\in \mathbb{N}_{0}}\left | \left \langle f,\psi_{l,j,k} \right \rangle \right |^{2}=\left \| f \right \|^{2}, \forall f\in L^{2}(\mathbb{K}) $

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Theorem. Let $X(\Psi)$ be Parseval wavelet frame. then every $f \in L^{2}(\mathbb{K})$ can be written as

$f(x)=\sum_{l=1}^{L}\sum_{j\in \mathbb{Z}}\sum_{k\in \mathbb{N}_{0}}\left \langle f,\psi_{l,j,k} \right \rangle\psi_{l,j,k}(x)$

Answer 1

This is a property that holds for any Parseval frame. The following is just a special case:

Note that, for all $f \in L^2 (\mathbb{K})$, $$ \langle f, f \rangle = \sum_{l=1}^{L}\sum_{j\in \mathbb{Z}}\sum_{k\in \mathbb{N}_{0}} \langle f,\psi_{l,j,k} \rangle \langle \psi_{l,j,k},f \rangle \ $$ whenever the wavelet system $X (\Psi)$ forms a Parseval frame for $L^2 (\mathbb{K})$. Therefore, $$ 0 = \langle f, f \rangle - \sum_{l=1}^{L}\sum_{j\in \mathbb{Z}}\sum_{k\in \mathbb{N}_{0}} \langle f,\psi_{l,j,k} \rangle \langle \psi_{l,j,k},f \rangle \ = \bigg\langle f - \sum_{l=1}^{L}\sum_{j\in \mathbb{Z}}\sum_{k\in \mathbb{N}_{0}} \langle f,\psi_{l,j,k} \rangle \psi_{l,j,k}, \;f \bigg\rangle $$ for all $f \in L^2 (\mathbb{K})$, and hence $$ f = \sum_{l=1}^{L}\sum_{j\in \mathbb{Z}}\sum_{k\in \mathbb{N}_{0}} \langle f,\psi_{l,j,k} \rangle \psi_{l,j,k}. $$