So, I'm currently taking a module in linear partial differential equations, and in said module we're taught much about the calculus (or theory) of distributions. In particular, we're taught about the convolution of two distributions, and at one point we are shown a property of the convolution operator which states the following:
If $f, g\in\mathcal{D}'(\mathbb{R}^N)$ are such that $f\ast g$ exists, then, $\forall h\in\mathbb{R}^N$, $f(x + h)\ast g(x) = (f\ast g)(x + h)$.
And I'm really quite confused as to what exactly the notation here means to imply. In particular, since the convolution of two distributions is defined in terms of the tensor product of two distributions, the expression $f(x + h)\ast g(x)$ makes sense to me, since, intuitively, you just "broadcast" the arguments of $f$ and $g$ to their respective representations in the tensor product of $f(x + h)$ and $g(x)$ as follows: $\langle f(x + h)\ast g(x), \varphi(x)\rangle = \langle f(x + h)\otimes g(x), \varphi(2x)\rangle$ (like I said, this is an intuitive expansion of the expression $f(x + h)\ast g(x)$, meaning that I don't actually know if it's correct). However, I'm really not sure what to make of the expression $(f\ast g)(x + h)$, probably because there's only one argument here, and thus I'm not particularly sure where to broadcast said argument to in the expansion of $(f\ast g)(x + h)$ into its corresponding tensor product expression.
Any help in this regard would be greatly appreciated. Also please do not hesitate to ask for any clarification as far as notation is concerned, I understand that this field is rather notation-centric, so I have no problem with clarifying what is meant by the symbols seen in this question.