Oscillation of Fourier integrals with a
spectral gap.

Abstract:
Suppose that in a real Fourier series the first $m$ terms vanish.
The classical Sturm-Hurwitz theorem then claims that its 
sum has at least $2m$
zeros on an interval $|x|\le \pi$.
We prove an analogous result for the density of sign changes
for sufficiently good functions with a spectral gap (joint
work with Eremenko).