Is there a smooth, non-zero $2\pi$-periodic function $f,$ with support of $f$ contained in an interval $[a,b]\subset[0,2\pi],$ such that $b-a<2\pi$ and only finitely many Fourier coefficients of $f,$ $$\hat{f}(n)=\int_0^{2\pi}f(x)e^{inx}dx$$ are non-zero? Explain your answer.
My attempt: The fourier expansion has the form $$\frac{a_0}{2}+\sum_{n=1}^Na_n\sin(nx)+\sum_{n=1}^Nb_n\cos(nx).$$ Probably we could take points where the function has values $0$(there are infinitely many points) to conclude that all Fourier coefficients are $0$?
Also what puzzles me is that how could a periodic function defined on $\mathbb{R}$ only have a bounded support. Can anyone help me please?
HINT. Use the exponential formulation of Fourier series. You are trying to prove that, if $$ f(x)=\sum_{-N}^N c_n e^{i n x} $$ vanishes at an infinite number of points, then $f$ is identically zero. You are doing well.
To conclude, try a clever substitution that makes $f$ into (the restriction to the unit circle of) a complex polynomial.