I'm working through Damelin and Miller's The Mathematics of Signal Processing for class and there is a paragraph about the Discrete Fourier Transform which simply defeats me:
How can a Fourier coefficient be 'projected'? And against what basis? I'm really having trouble understanding this and would greatly appreciate any help.
I'm not sure what the Euclidean algorithm has to do with this, other than also using long integer division/division-with-remainder.
The projection is not of the single coefficient, but of the full sequence $\{\hat f(a)\}_{a\in\Bbb Z}$ to the finite sequence $$ a\mapsto \sum_{b\in\Bbb Z}\hat f(a+bN), ~~~ a=0,1...,N-1. $$ To make this a true projection, one would have to extend this finite sequence to a doubly infinite sequence by zero-padding. Then applying the summation-projection again, each sum has only one non-trivial term, so that indeed the square of the operator is the operator itself.