Oscillation of functions with a spectral gap.
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- Record sourced from PubMed, PMID 15079072.
- Also identified by PMC identifier 395890.
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Abstract
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.