I run in to a real problem which must be a classic, only I cannot find the answer. I know Fourier transform shifts a signal from the time spectrum to the frequency spectrum. Is there a similar transform to shift a function from the time spectrum to the phase spectrum?
I.e., where Fourier translates $g(t) \rightarrow a_0 / 2 \sum_1^\infty a_n \cos(nt) +b_n \sin(nt)$ I am wondering if there is a likewise transform that translates $g(t) \rightarrow a_0 + \sum_1^\infty a_n \cos(t+2\pi/n) +b_n \sin(t+2\pi/n)$? Or anything of the likes of that?