[qppo]

I mean ultimately it comes from functional analysis and differential equations (not signal processing).

It's a binary operator on functions that yields a third function. It has a lot of useful properties and equivalences, like that it can be described as the product of two Fourier transforms (although that's very roundabout).

You're actually introduced to convolution in middle school when you're taught how to multiply monomials to build a polynomial (at my middle school they called it "FOIL").

[elcritch]

It appears to be a discrete Fourier, no? Does it apply to all convolutions or just a specific instance or subset? As in id there a proof showing that as sample size N goes to a limit it approaches a continuous limit? I still natively think in continuous convolutions from Physics. The whole discretization of these operators is oddly harder for me despite it technically being simpler to compute.

[qppo]

No, it's true of both continuous and discrete time/domain Fourier. Convolution in time is multiplication in frequency, and vice versa. You don't need to prove this with limits directly, just use the definition of the convolution integral and Fourier transform integral.

> technically being simpler to compute.

They're equivalent, since the only meaningful way to "compute" a continuous convolution is symbolically, and discrete convolutions obey most of the same identities.

If one can place a lower bound on the time step resolution of a simulation then continuous convolutions are evaluated using discrete convolutions, which can represent the continuous case exactly via the Nyquist-Shannon sampling theorem.

Interestingly enough, to prove the Sampling Theorem you need to rely on the identity that multiplication in frequency is convolution in time, and to prove that it can't be realized in a physical system (breaks causality, since you multiply by a superposition of Heavisides which of course are infinitely long sinc functions in both directions of time).

And more interesting is that signals and systems is mostly applied dynamics and statistics, so it shouldn't be surprising if there's overlap.