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Wavelet basis are constructed applying translations and dilatations in appropriate functions called mother wavelets. For discrete wavelet transformations, for example, one can create an orthonormal basis using the transformations $$\Psi_{jk}(x) = 2^{-j/2}\Psi(2^j x +k),$$ where $\Psi$ is some mother wavelet function and $j,k\in\mathbb{Z}$. There is some way to construct an orthonormal basis for a continuous wavelet transformation like that one below $$\Psi_{\tau s}(x) = \tau^{1/2}\Psi\left(\frac{x+s}{\tau}\right),$$ where $\tau$ and $s$ are real parameters and $\Psi$ is supported in $\mathbb{R}$?

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Observe that $\tau\mapsto \Psi_{\tau s}$ is continuous with respect to the usual metrics on $\mathbb{R}$ and $L^{2}(\mathbb{R})$. Then if $\{\Psi_{\tau s}\}_{\tau\in\mathbb{R}}$ were an orthonormal collection, we would have $0=\langle \Psi_{\tau s},\Psi_{\tau' s}\rangle\rightarrow \langle \Psi_{\tau s},\Psi_{\tau s}\rangle=1$ as $\tau'\rightarrow\tau,$ which is clearly impossible.

Clearly the collection $\{\Psi_{\tau s}\}_{\tau,s\in\mathbb{R}}$ may be sufficient to represent all of $L^{2}(\mathbb{R})$ with linear combinations thereof, but this representation will have many more functions than is necessary, unlike in a basis, where we have in some sense a minimal representation.

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