Admissibility condition for wavelets

Remarks

If $\psi$ is also in $L^1(\R)$, then $\hat\psi$ is continuous, and admissibility implies $\hat\psi(0) = \int_\R \psi(t) \, dt = 0$. Conversely, if $\int_\R (1 + \abs{t}) \abs{\psi(t)} \, dt < \infty$ and $\int_\R \psi = 0$, then $\hat\psi$ is Lipschitz with $\hat\psi(0) = 0$, and $\psi$ is admissible. The constant $C_\psi$ is the factor in Calderón's identity for the continuous wavelet transform.

Example: the Mexican hat $\psi(t) = (1 - t^2) e^{-t^2/2}$ is admissible. It decays fast and $\int_\R \psi = \int_\R e^{-t^2/2} \, dt - \int_\R t^2 e^{-t^2/2} \, dt = \sqrt{2\pi} - \sqrt{2\pi} = 0$, so the converse criterion applies.

Non-example: the Gaussian $\psi(t) = e^{-\pi t^2}$ has $\hat\psi(\xi) = e^{-\pi \xi^2}$. Near $\xi = 0$ the integrand behaves like $1/\abs{\xi}$, which is not integrable, so $C_\psi = \infty$.