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?
Best Answer
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.