Determine the domain of $(f\oplus g)$ in terms of the domains of $f$ and $g$

Question

I am having trouble understanding convolution and would like some help with the following:

Let f and g be functions from $X$ to $]-\infty,+\infty]$. Determine the domain of $(f\oplus g)$ in terms of the domains of $f$ and $g$.

I have the following definition

$$(f\oplus g)(z)=\inf_{x+y=z}(f(x)+g(y))$$ however, I am not sure how this may help me. Any suggestion will help me greatly.

Answer 1

I'm going to make some assumptions here that you probably should have made somewhat explicit in your question. That is, I will assume that $X$ is a vector space, and that $\mathop{\textrm{dom}} f, \mathop{\textrm{dom}} g\subseteq X$.

What that in mind, the quantity $f(x)+g(z-x)$ is defined only when $x\in\mathop{\textrm{dom}} f$ and $z-x\in\mathop{\textrm{dom}} g$, which means that $z\in\mathop{\textrm{dom}} f+\mathop{\textrm{dom}} g$. That is,

$$\mathop{\textrm{dom}} (f\oplus g) = \mathop{\textrm{dom}} f+\mathop{\textrm{dom}} g = \left\{z\in X \,|\, \exists x\in\mathop{\textrm{dom}} f, y\in\mathop{\textrm{dom}} g ~\text{s.t.}~x+y=z\right\}$$

Answer 2

You can rewrite

$$(f \oplus g)(z) = \inf_{x\in \mathbb X} (f(x)+g(z-x))$$

So $(f\oplus g)(z)$ is only defined if $x \in X$ and $z-x \in X$, which is equivalent to $x \in X \cap (z-X)$, where $z-X := \{z-y |y \in X\}$