Oscillation of functions with a spectral gap.
We prove an old conjecture on oscillation of functions that have a spectral gap at the origin. Suppose that the Fourier transform of a real measure f on the real line satisfies f(x) = 0 for x in (-a, a). Then, when r --> infinity, the asymptotic lower density of the sequence of sign changes of f on the intervals [0, r) is at least a/pi. This still holds for some wider classes of measures characterized by their rate of growth at infinity, but if the growth is faster than a certain threshold, the above statement is no longer true.
Eremenko
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Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. eremenko@math.purdue.edu
Novikov
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