[thearn4]

I've come to internalize the Fourier transform as a type of sparse matrix algorithm. I.e. something that implements a particular matrix-vector product without requiring explicit construction of said matrix.

[srean]

And pushing the parantheses to factor out common subexpressions. Essentially exploiting the distributive law

[enchiridion]

Do you mean the FFT? I've been trying to wrap my head around the Fourier transform lately, and I can see the matrix connection to an FFT, but not the Fourier transform in general.

[bonoboTP]

The Fourier transform performs a change of basis into the eigenbasis of convolution.

It's linear and so in the discrete case can be expressed as matrix multiplication (every discrete linear operation is expressible as matmul).