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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?

carlosayam
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1 Answers1

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I have to agree with Jack Peetre, who remarked at the end of his review of Calderón's pioneering paper:

The presentation is not very clear ... the reader has to spend quite a lot of time just searching for the relevant passage for the proof of each particular statement.

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.

A modern exposition of this condition appears in The Mathematical Theory of Wavelets by Weiss and Wilson: see Theorem 2.1.