Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on frequency analysis compared to the Fourier transform. I've always wondered about possible generalizations of the Fourier transform. For instance, I posted a related question on MO: Fourier series but different waveform.

An interesting question arises regarding whether the CWT is injective. Although I'm not well-versed in the theory of Continuous Wavelet Transform, I've made some attempts to analyze this question.

For $a \neq 0, b \in \mathbb{R}$, let $f_{ab}(x)=f(a(x-b))$ and $f_a(x)=f(ax)$ for a function $f$. Below are some statements:

1. Let $0 \neq \tau \in \mathcal{D'}(\mathbb{R})$ and $f \in \mathcal{D}(\mathbb{R})$. If $\tau(f_{ab}) = 0$ for all $a, b$, then $f=0$.
2. Let $0 \neq \tau \in \mathcal{S'}(\mathbb{R})$ and $f \in \mathcal{S}(\mathbb{R})$. If $\tau(f_{ab}) = 0$ for all $a, b$, then $f=0$.
3. Let $0 \neq g \in L^2(\mathbb{R})$ and $f \in L^2(\mathbb{R})$. If $\int gf_{ab} = 0$ for all $a, b$ (equivalently, if $\int g_{ab}f = 0$ for all $a, b$ ), then $f=0$. This implies that the dilations and translations of a nonzero $g \in L^2(\mathbb{R})$ are dense in $L^2$.

Here is my progress in verifying them. Let's first consider Statement 2. Note that $\tau(f_{ab})=0$ for all $a,b$ implies $\tau * f_a = 0$ for all $a$. By applying the convolution theorem of the Fourier transform, this is equivalent to saying:

4. Let $0 \neq \tau \in \mathcal{S'}(\mathbb{R})$ and $f \in \mathcal{S}(\mathbb{R})$. If the tempered distribution $\tau f_a = 0$ for all $a \neq 0$, then $f=0$.

For the same reason, Statement 3 is equivalent to:

5. Let $0 \neq g \in L^2(\mathbb{R})$ and $f \in L^2(\mathbb{R})$. If $gf_a = 0$ almost everywhere for all $a\neq 0$, then $f=0$ almost everywhere.

Statement 4 seems promising, but it is false. A counterexample is letting $\tau$ be the Dirac function and $f$ be a function supported in $\{x>0\}$. Consequently, Statement 2 is also false.

We can directly find a counterexample for Statement 2: let $\tau$ be the constant function $1$ and $f$ be of zero mean. This is also a counterexample for Statement 1.

However, Statement 5 (and also Statement 3) appear to be true. Firstly, notice that in Statement 3, we can assume $f \in \mathcal{S} (\mathbb{R})$ without loss of generality. By applying a Fourier transform, we can also assume in Statement 5 that $f$ is in $\mathcal{S}(\mathbb{R})$, which then makes Statement 5 obvious from the continuity of $f$.

Here is a related question I asked earlier on MO: Multiplication with dilations of nonzero measurable function is injective.

I have the following questions:

• Which conditions can we impose on $\tau$ to make Statements 1 and 2 true? I speculate that the condition $\operatorname{supp} \tau \not\subset \{0\}$ would suffice for Statement 4 to be true, although I don't know how to prove it rigorously. For Statement 1, we cannot use the Fourier transform trick, and I'm unsure how to address this. A notable example for Statement 2 is letting $\tau = \exp(i\cdot)$, which corresponds to the familiar Fourier transform.
• Can we prove Statement 3 without involving the Fourier transform? Are dilations and translations still dense in $L^p$ for other $p \geq 1$? I find this to be a fascinating question in real analysis, but I haven't encountered it in any textbooks.
• Are there references on Continuous Wavelet Transform that focus on the theoretical part? What are the conditions for the injectivity to hold? Are there theories not only for $L^2$, but for more general functions, like the theory of Fourier transform for tempered distributions? It seems that many books on Wavelet Theory focus on its applications in engineering. To derive elegant formulas and algorithms, they often impose specific conditions on the functions $f, g$ in Statement 3. For example, there is Morlet's wavelet reconstruction formula for CWT, as discussed in this MSE question: Morlet's wavelet reconstruction formula. It appears that there is an integrability condition (the finiteness of $B_\psi$) for the formula to be well-defined.