Uniqueness for the continuous wavelet transform (Q2910615)

In analogy with Fourier transforms, uniqueness of a wavelet transform is defined as a condition that if the continuous wavelet transform of a function (in a certain class) vanishes identically then the function is zero (as an element of the given class). The wavelet transform \((W_\psi f)(s,t)\) of \(f\) is defined here as NEWLINE\[NEWLINE (W_\psi f)(s,t)=\langle f,\psi_{s,t}\rangle =\int_{\mathbb{R}^d} f(x)\, \overline{\psi_{s,t}(x)}\, dx\, s>0,\, t\in \mathbb{R}^d,NEWLINE\]NEWLINE where \(\psi_{s,t}(x)=\psi((x-t)/s)/s^{d/2}\). The first proposition is a spherical-polar conversion and states that if \(\psi\in L^2(\mathbb{R}^d)\) and if \( (W_\psi f)(s,t)\) vanishes identically then for almost every unit vector \(\xi'\) one has NEWLINE\[NEWLINE\int_0^\infty |\hat{f}(r\xi')|^2\, r^{d-1}\, dr \int_0^\infty |\hat{\psi}(s\xi')|^2 s^{d-1}\, ds =0\, . NEWLINE\]NEWLINE Here {\textit{almost every}} can be taken with respect to surface measure on the unit sphere.NEWLINENEWLINEThe first main theorem states that if \(\psi\in L^{p'}(\mathbb{R}^d)\) has Fourier transform smooth away from the origin, and if \(f\in L^p(\mathbb{R}^d)\) (\(1/p+1/p'=1\)) is such that \((W_\psi f)(s,t)\) vanishes identically then \(f=0\) a.e.~if \(\hat\psi\) is {\textit{nontrivial}} in every direction, meaning that \(\widehat\psi(r\xi')\neq 0\) on a set of \(r\in (0,\infty)\) of positive measure for each \(\xi'\).NEWLINENEWLINEA second main result considers functions \(f\) such that \(f(x)(1+|x|)^{-k}\in L^\infty(\mathbb{R}^d)\) for some \(k\in \mathbb{N}\). One supposes now that \(\psi(x)(1+|x|)^k\in L^1(\mathbb{R}^d)\) and \(\hat\psi\) again is smooth away from the origin and nontrivial in almost every direction. Then the condition that \((W_\psi f)(s,t)\) vanishes identically implies that \(f\) is a polynomial of degree at most \(k\). The main technical tool is a Calderón-type reproducing formula adjusted to the hypotheses.