Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true:
Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost everywhere equal to $0$. If $gf_s$ is almost everywhere equal to $0$ for every $s \in \mathbb R \setminus \{0\}$, then $g$ must be equal to $0$ almost everywhere.
Obviously, this statement must be true if $f$ or $g$ is continuous. I don't know whether it still holds in the general measurable case.
This might seem to be a meaningless question, but it has background. In fact, if $f, g \in L^2$, then from the convolution theorem and $L^2$ theory of Fourier Transform, there is an equivalent statement:
Suppose $f, g \in L^2(\mathbb{R})$, and $f \neq 0$ in $L^2$. If $g*f_s$ is $0$ for every $s \in \mathbb R\setminus \{0\}$, then $g = 0$.
I believe this statement concerning convolution with dilations is equivalent to the injectivity of the Continuous Wavelet Transform:
Suppose $f, g \in L^2(\mathbb{R})$, and $f \neq 0$ in $L^2$. If for every $s\in\mathbb R\setminus \{0\}, b\in \mathbb R$, $$\int g(x)f(s(x-b))\,dx=0.$$ Then $g$ must be $0$.
I think this is a rather interesting and promising question. If $f=\chi_{[0,1]}$, then it is merely saying that if a function $g$ has zero average on every interval, then $g$ must be zero.
For the Continuous Wavelet Transform, there is a Morlet's wavelet reconstruction formula, as shown in this MSE question: Morlet's wavelet reconstruction formula. However, there seems to be an integrability condition (the finiteness of $B_\psi$) for the formula to be well-defined. I wonder whether the Continuous Wavelet Transform is still injective in the more general case.
I am not familiar with the theories of wavelets. Please correct me if there is any mistake.
Edit: As KhashF showed in the comment, this statment is false if $s$ is only considered to be positive. So I have changed "$s > 0$" to "$s \in \mathbb R\setminus \{0\}$". This really confuses me, because in the wiki of Continuous Wavelet Transform and the referenced MSE question, only positive dilations are considered, but then it won't be injective in most cases, as the convolution theorem shows. I guess they have made the assumption that the mother wavelet $\psi$ is real, so its Fourier transform is conjugate symmetric.
I have also corrected the question as Ben Johnsrude suggested.
