We know that, If $f$ and $g$ are functions then their convolution is defined as,
$(f*g)(x) =\int_{-∞}^{∞} f(t)g(x-t) dt$
(This is the convolution structure for Fourier transform)
But, what if $f$ and $g$ are functions of two variables? say $f(x, y)$ & $g(x, y)$ then how we can define the convolution in this case? (Please provide reference too)
Update: I have seen many text where it is defined like this, $$(f*g)(x,y) =\int_{-∞}^{∞}\int_{-∞}^{∞} f(t, s)g(x-t, y-s) dtds$$.
But I am unable to understand how they arrive at this definition.
Please help me
It is defined entirely analogously: Given two integrable functions $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}^n$, one defines $$(f*g)(x) := \int\limits_{\mathbb{R}^n} f(y)g(x-y) dy $$
for all $x\in\mathbb{R}^n$. As a reference, see e.g. Convolution.
Concerning the existence of $(f*g) : \mathbb{R}^n\longrightarrow\mathbb{R}^n$, there are several criteria based on the input functions $f$ and $g$. The easiest one being that if $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$, then $(f*g)\in L^p(\mathbb{R}^n)$ due to Young's inequality.