# Is there a 3D version of the convolution and cross-correlation theorems for the Laplace transform?

The Laplace-transform version of the convolution and cross-correlation theorems is essentially the same as the "usual" (Fourier-transform) version: if $$\mathcal{L}[f(t)]$$ is the Laplace transform of a function $$f(t)$$, given by $$\mathcal{L}[f(t)]=\int_0^\infty f(t)\,e^{-st}\,\text{d}t,$$ and if $$f\ast g$$ is the convolution of $$f$$ and $$g$$ and $$f★ g$$ is the cross-correlation of $$f$$ and $$g$$, given by $$(f\ast g)(t)=\int_0^t f(\tau)\,g(t-\tau)\,\text{d}\tau$$ and $$(f\star g)(t)=\int_0^\infty f^\ast(\tau)\,g(t+\tau)\,\text{d}\tau,$$ then $$\mathcal{L}[(f\ast g)(t)]=\mathcal{L}[f(t)]\,\mathcal{L}[g(t)]$$ and $$\mathcal{L}[(f\star g)(t)]=\mathcal{L}[f^\ast(-t^\ast)]\,\mathcal{L}[g(t)],$$ where $$^\ast$$ denotes complex conjugation.

I'd like to know whether those also hold for functions of multiple variables (e.g. $$f(t_1,t_2,t_3)$$ and $$g(t_1,t_2,t_3)$$) and, if they do, whether any of the usual shenanigans involving higher-dimensional calculus occur here (e.g. whether the argument of $$g$$ in the convolution integral is $$g(t_1-\tau_1,t_2-\tau_2,t_3-\tau_3)$$ or $$g(|\vec{t}-\vec{\tau}|)$$ with $$\vec{t}=(t_1,t_2,t_3),\vec{\tau}=(\tau_1,\tau_2,\tau_3)$$ or something else, and similarly for the argument of $$g$$ in the correlation integral).

The Wikipedia article on the convolution theorem states the theorem for the Fourier transform (as is usual as far as I'm aware), says that "[t]he theorem also generally applies to multi-dimensional functions" and then (in a different section) states that "[t]his theorem also holds for the Laplace transform", but it doesn't specifically state that the generalisation to functions of multiple variables also holds for the Laplace transform. Intuition says it should, but when I tried my hand at a simple proof I ran into a wall.

Wikipedia has no article on the cross-correlation theorem, but it mentions the theorem on its article on the discrete Fourier transform has a section on the theorem (in which it doesn't mention the Laplace transform at all).

The Wikipedia article on the Laplace transform has a section in which it states both theorems for single-variable functions.

I haven't been able to find the required information, and especially the bit about the argument of $$g$$ in the integrals, anywhere.

• You should delete one of them. Commented Nov 15, 2023 at 19:27
• @Moo The double post is probably just due to a bug. I think it happened once recently? Commented Nov 15, 2023 at 19:35
• Sounds like a bug, or maybe I accidentally clicked "ask question" twice. Apologies if it was me; I've deleted the other one.
– Rain
Commented Nov 15, 2023 at 22:46