under what condition a periodic function will have finite Fourier-coefficient?

Question

for a periodic function f(x)(continuous & have continuous derivatives), there is a Fourier series for it, i just wonder under what condition f(x) will have finite(or infinite) Fourier-coefficent, which means existing N so kn=zero for any n>N.

and is there any significant difference in function shape between the functions of infinite and infinite spectrum?

Answer 1

$f(t)=\sum_{n=-N}^N c_n e^{i n t}$ iff $f$ is $2\pi$-periodic, smooth, and $D^{2N} (e^{iNt} f)=0$ where $D (g)(t) =e^{-it} g'(t) $.