I have two functions $n(x)$ and $g(x,y)$, and I would like to know if there is a way to express $p(x)$, given that the following relationships are true:
- $n(x)$ is the result of vertically integrating some 2d function $f(x,y)$, i.e. $n(x) = \int f(x,y)\;\textrm{d}y$
- $p(x)$ is the result of vertically integrating some 2d function $h(x,y)$, i.e. $p(x) = \int h(x,y)\;\textrm{d}y$
- the 2d functions $h(x,y),f(x,y)$, and $g(x,y)$ are linked by a convolution, i.e. $h(x,y) = f(x,y)\circledast g(x,y)$.
Schematically, the problem looks like the following:
I know that the convolution can be performed in Fourier space $$ \begin{align} h(x,y) &= f(x,y)\circledast g(x,y) \\ &= \mathcal{F}^{-1}\{ \mathcal{F}(f)\cdot\mathcal{F}(g) \}, \end{align} $$ but is there a way to write down an expression for $p(x)$, given the above integration relationships, and the fact that I only have access to $n(x)$ and $g(x,y)$?
I do not have access to $f(x,y)$.
Thank you!
---Edit---
From doing some numerical tests, it seems like the solution should be $$ p(x) = n(x) \circledast \int g(x,y)\;\textrm{d}y. $$ Is there a rule which says this can be done?