Nit to your nit, which is incorrect w.r.t. parent reply:
A Fourier transform is not a projection, it's a change of basis represented by a unitary transformation.
The w Fourier coefficient F(w) is the dot product of f with an exponential function, `e_w • f`, and is in that sense a projection. The inverse Fourier transform writes the original function as a sum of the projected components: `f = sum_w (e_w • f) e_w = sum_w F(w) e_w`. This is exactly how writing an "arrow" style 2- or 3-D vector as a sum of orthogonal projections works.
True, but missing the point as you would not normally refer to the identity matrix as a projection.
What was your point? An incorrect characterization of projections?