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I am trying to prove that the sytem $\{\psi_{j,k}(x)=2^{\frac{j}{2}}\psi(2^j x-k)\}_{j,k \in \mathbb{Z}}$ is an orthonormal basis of $L^2(\mathbb{R})$. I've managed to prove that it is an orthonormal system but I am stuck in the basis part.

I have been told that I have to use the fact that $L^2(\mathbb{R})=\oplus L^2 (S_j)$, where $S_j=(-2^j,-2^{j-1}] \cup [2^{j-1},2^j)$ are intervals that come up when proving $\{\psi_{j,k}\}_{j,k \in \mathbb{Z}}$ is an orthonormal system.

My approach is the following:

By Parseval's theorem, it is enough to prove that $\{(\psi_{j,k})\wedge\}_{j,k \in \mathbb{Z}}$ is an orthonormal basis. I have proved that $$(\psi_{j,k})\wedge(\xi)=2^{\frac{-j}{2}}e^{\frac{2\pi i \xi k}{2^j}} \chi_{[2^{j-1},2^j)}(|\xi|)$$

In the case $j=0$, we got $(-1,-\frac{1}{2}] \cup [\frac{1}{2},1)$, which can be viewed as $[0,1)$ as a traslation, and there we know that the exponentials are an orthonormal basis. For $j\geq 1$, it is again a traslation. My problem is with the interval $[-\frac{1}{2},\frac{1}{2}]$. I do not know hot to deal with it.

I would appreciate any help.

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