Thanks Professor. I have read the paper but the paper proved a lower bound of the sign changes density asymptotically, if I understand correctly. So it does not deal with the case that $\hat f$ is supported on $(-M, -1)\cup (1, M)$.

Sorry I made a typo. I meant that $f$ has Fourier coefficients supported on $(-M, -1)\cup (1, M)$. Will the corresponding conclusion change? Thanks!

Thank you very much! I have a further question. If a real-valued function $f\in L^2$ has Fourier coefficients vanishing on $(-M, -1)\cup (1, M)$, where $M>1$, is it guaranteed that $f$ has a sign change in $(0, cM)$?

Thanks! Locality is a bad word here. Here I mean the function is monotone on [0, cN] for some universal constant c > 0.