Values of Lebesgue integrable function from the integral of the product of a known function.

Question

Let's say we have three complex absolutely integrable and square integrable functions $A,B,C\in \mathbb{L}^1(\mathbb{C})\cup \mathbb{L}^2(\mathbb{C})$ such that the following holds: $$A(y) = \int_{-\infty}^\infty{B(x)C(xy)dx}$$ If we can measure or calculate $A$ and $C$ for all values, what information (if any) could we learn about the function $B$? I understand that the entire function $B$ can't be extracted in most cases, but is there a systematic way of describing what information can be extracted and how to do so?

Answer 1

There is an algorithm similar to what I was looking for in this physics paper. It uses the de-convolution theorem and polynomial series to "invert" the integral. I use the term "invert" loosely here because information is definitely lost, but for some functions it clearly produces decent approximations.