# Calderon's admissibility condition for wavelets explained

Calderon's admissibility condition is a central argument in a number of recent wavelet-like constructs, like curvelets, shearlets, to name a few. It states that if $\psi$'s Fourier transform conforms to :

$$\int_0^\infty \left| \hat{\psi}(a \xi )\right| ^2\frac{da}{a}=1, \ \ \ \forall \xi \in R$$

then $\psi$ can be used as a wavelet because it generates a tight frame decomposition.

Authors point to this paper: Calderon, "Intermediate spaces and interpolation, the complex method", Studia Mathematica, T. XXIV (1964). I read the paper but couldn't get where the admissibility condition was proved.

Could anyone point me to another proof?

-

Calderón's reproducing formula appears at the top of page 128 in the form "it is possible to select $\psi_1$ and $\psi_2$ in such a way that $\mathscr{S}(Tu,u)=u$", where $\mathscr{S}$ and $T$ are integral operators defined on pages 126-127. Here $T$ is a prototype of wavelet transform and $\mathscr{S}$ the corresponding reproducing operator. The conditions (5)-(6) on page 184 appear to be a prototype of the integral condition in your post.