Admissibility condition for wavelets The admissibility condition for a wavelet $\psi$ is:
$\int \frac{|\hat\psi(x)|^2}{|x|} dx < \infty$, with $\hat\psi$ the Fourier transform of $\psi$.
A necessary and sufficient condition should be:
$$\int \psi(x) dx = 0\quad \mathrm{(or} \;\hat\psi(0) = 0\mathrm{)}$$  
But I can't see how you should get from one to the other...
Does someone have some suggestions?
 A: The admissibility condition (at least in the most common versions in 1D) can be stated as follows: function $\phi$ is a synthesis (or reconstruction) wavelet for analysis wavelet $\psi$ if the two constants:
$$c^{\pm}_{\phi,\psi} =
2\pi \int_{0}^{+\infty}\tfrac{\hat{\overline{\psi}}(\pm\omega)\hat\phi(\pm\omega)}{\omega} d\omega
$$
 are finite, non null and equal (see M. Holschneider, Wavelets, an analysis tool, 1995, p. 65 sq.).
This can be satisfied by a lot of couples of functions. If you now want the synthesis wavelet to be the same as the analysis wavelet, ie $\phi = \psi$, then a necessary condition is that $$\int \psi = 0$$ to avoid the singularity at $0$. This is the root for using discrete orthogonal wavelets (otherwise, you have for instance biorthogonal wavelets, used in JPEG 2000 image compression). In other words, the wavelet has zero-mean, which is equivalent to say (when integrals exists) that its spectrum vanished at zero (on the frequency axis).
But it is not sufficient in theory, and requires additional smoothed/boundedness/localization on $\psi$. So my suggestion is to refer to accurate sources for sufficient conditions.
