G(\omega) \, e^{i\omega t} d\omega
\end{multline}`

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#The Wavelet Transform

We see that the wavelet transform  is the filtration of `$z(t)$` with  rescaled versions of  `$\psi(t)$` using the cross-correlation operator:

`\[w(t,s) =\int_{-\infty}^\infty \frac{1}{s} \psi^*\left(\frac{\tau-t}{s}\right) z(\tau) d\tau\]`

From the cross-correlation theorem and scaling theorems, we have

`\[w(t,s) =\int_{-\infty}^\infty \frac{1}{s} \psi^*\left(\frac{\tau-t}{s}\righ