Definition of Complex Convolution?

Question

The convolution of two real value functions f and g is defined to be
$$(f \ast g)(t) = \int_{-\infty}^\infty f(\tau)g(t-\tau)\,d\tau$$ If I extend this idea into higher dimension (Let's say in a unit disc or maybe $ \mathbb{C}$, a set of complex number), can we defined a convolution of two complex value function?
My guess is going to be a double integral in some way. Could you please precisely state it?

Answer 1

The convolution of functions has a natural and very important generalization to functions defined on any locally compact topological group (such as $\mathbb C$, $\mathbb R^n$, the circle, $\text{GL}_n$, $\text{SL}_n$, $\text{O}_n$, $\text{U}_n$, $\mathbb Z$, $\mathbb Z_n$, any discrete group, and many many others).

If the group is called $G$, then the convolution is defined by $$ (f * g)(x) = \int_Gf(y)g(y^{-1} x) \, d\mu(y), $$ whrere $\mu $ is the Haar measure on $G$, namely the unique (up to a scalar multiple) left-invariant measure on $G$.

To be more precise, in the case of

• $G=\mathbb C$ $$ (f * g)(x+iy) = \int_{-\infty}^\infty \int_{-\infty}^\infty f(z+iw)g(x+iy - z-iw) \, dzdw, $$

• $G=\mathbb T = \{z\in \mathbb C: |z|=1\}$ $$ (f * g)(e^{i\theta }) = \frac 1{2\pi}\int_{0}^{2\pi } f(e^{i\tau })g(e^{i(\theta -\tau )}) \, d\tau , $$

• $G=\mathbb Z$ $$ (f * g)(n) = \sum_{k=-\infty }^{\infty } f(k)g(n-k). $$

Finally, since the unit disk in the complex plane is not a group, there is no natural convolution for functions defined there, unless you extend these functions to the whole plane by setting them to be zero outside the disk.

Answer 2

I believe the result of Fourier convolution defined in formula (1) below is only valid for $t\in\mathbb{R}$, whereas the result of a Mellin convolution such as the two defined in formulas (2) and (3) below can also be valid for at least a subset of the complexes such as $\Re(s)>0$.

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(1) $\quad (f \ast g)(t)=\int\limits_{-\infty}^\infty f(\tau)\,g(t-\tau)\,d\tau$

(2) $\quad (f \ast_{M_1} g)(t)=\int\limits_0^\infty f(x)\,g\left(\frac{y}{x}\right) \frac{dx}{x}$

(3) $\quad (f \ast_{M_2} g)(t)=\int\limits_0^\infty f(x)\,g(y\,x)\,dx$

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For examples, see formulas (52) - (70) in this answer I posted to one of my own questions on Math StackExchange.