I have some problems understanding a couple of proofs about wavelets, all theese proofs have the same computation i do not understand (i can not see why it is true). Firts, i give you some context. Consider the following definition:
We say a family $\{V_{j}\}_{j \in \mathbb{Z}}$ of closed subspaces in $L^{2}(\mathbb{R})$ is a multiresolution analysis (MRA) if satisfies the following:
- $V_{j} \subset V_{j-1}$
- $\bigcap_{j \in \mathbb{Z}}V_{j} = \{0\}$
- $\overline{\bigcup_{j\in\mathbb{Z}}V_{j}} = L^{2}(\mathbb{R})$
- $f(x)\in V_{j} \Leftrightarrow f(2^{j}x)\in V_{0}$
- $f(x)\in V_{0} \Rightarrow f(x-n)\in V_{0}$ para todo $n \in \mathbb{Z}$
- There exists a function $\phi \in V_{0}$ such that $\{\phi_{0,m}\}_{m \in \mathbb{Z}}$ is an orthonormal basis for $V_{0}$
where $\phi_{j,k}(x) := 2^{-j/2}\phi(2^{-j}x-k)$. Using these conditions we can obtain that in a MRA the set $\{\phi_{j,m}\}_{m \in \mathbb{Z}}$ is an orthonormal basis for $V_{j}$. Because $\phi \in V_{0} \subset V_{-1}$ we can write:
$$\phi(x) = \sum_{m \in \mathbb{Z}}{h_{m}\phi_{-1,m}(x)} = \sum_{m \in \mathbb{Z}}{\sqrt{2}\cdot h_{m}\phi(2x-m)}$$
The computation i have problems with is obtained by taking the fourier transform on last equation:
$$\mathcal{F}\{\phi\}(\xi) = \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}{\sum_{m \in \mathbb{Z}}{\sqrt{2}\cdot h_{m}\phi(2x-m)}e^{-ix\xi}dx}$$
here, the book i am reading (Ten lectures on waveles - Ingrid Daubechies - ch.5 eq.5.1.16) interchange summation and integration. I have tried to use clasical theorems from measure theory: dominated convergence and monotone convergence theorem, but got nothing, just can not find the apropiate sequence that dominate this one. I guess there is something easy that i can not see, i am stucked a couple of days on this. I would apreciate any help.
If $f=\sum f_n$ in $L^{2}$ norm then $\mathcal F (f)= \sum \mathcal F (f_n)$ because the partial sums converge in $L^{2}$ and FT is continuous w.r.t. $L^{2}$ norm.