Event № 2173

 It is known that the Fourier transform of a measure which vanishes onIt is known that the Fourier transform of a measure which vanishes on
[-a,a] must have asymptotically at least a/pi zeroes per unit interval. One
way to quantify this further is using a probabilistic model: Let f be a
Gaussian stationary process on R whose spectral measure vanishes on [-a,a].
What is the probability that it has no zeroes on an interval of length L?
Our main result shows that this probability is at most e^{-c a^2 L^2},
where c>0 is an absolute constant. This settles a question which was open
for a while in the theory of Gaussian processes.I will explain how to
translate the probabilistic problem to a problem of minimizing weighted L^2
norms of polynomials against the spectral measure, and how we solve it
using tools from harmonic and complex analysis. Time permitting, I will
discuss lower bounds.
Based on a joint work with Ohad Feldheim, Benjamin Jaye, Fedor Nazarov and
Shahaf Nitzan (arXiv:1801.10392).