Orthonormal basis for continuous wavelet transformation

Question

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 $\mathbb{R}$?

Answer 1

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.