Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the [wiki][$1]). 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](https://mathoverflow.net/questions/467201/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](https://mathoverflow.net/questions/468638/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](https://math.stackexchange.com/questions/579199/morlets-wavelet-reconstruction-formula). It appears that there is an integrability condition (the finiteness of $B_\psi$) for the formula to be well-defined.

[$1]: https://en.wikipedia.org/wiki/Continuous_wavelet_transform