Edit: I denote ***, **, * for 3D, 2D, 1D convolution, respectively.
Let's denote an operator $R$ as
$$R[f](x,z) = \int_{-\infty}^\infty f(x,y,z) dy \tag{1}$$
where $f$ has compact support. Then, I want to evaluate
$$R[f \ast\ast\ast g](x,z) = \int_{-\infty}^\infty (f \ast\ast\ast g) (x,y,z) \,dy$$
- I wonder if $R[f \ast\ast\ast g] = R[f] \ast\ast R[g]$ holds. $R[f]$ becomes 2-dimensional function. I feel that it doesn't hold in general, but I'm not sure.
- Or wonder if it can be simplified if $g$ is a well-known function like Gaussian.
Context: In 2D, $R$ is similar to 2D Radon transform for a fixed angle and it is known that $\text{Radon}[f*\ast g]=\text{Radon}[f]*\text{Radon}[g]$. But I'm not sure about the case in (1) where $R$ is not Radon transform, but involved with a line integral.