PT - JOURNAL ARTICLE
AU - Eremenko, A.
AU - Novikov, D.
TI - Oscillation of functions with a spectral gap
AID - 10.1073/pnas.0302874101
DP - 2004 Apr 20
TA - Proceedings of the National Academy of Sciences of the United States of America
PG - 5872--5873
VI - 101
IP - 16
4099 - http://www.pnas.org/content/101/16/5872.short
4100 - http://www.pnas.org/content/101/16/5872.full
SO - Proc Natl Acad Sci U S A2004 Apr 20; 101
AB - 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 f̂(x) = 0 for x ∈ (–a, a). Then, when r → ∞, the asymptotic lower density of the sequence of sign changes of f on the intervals [0, r) is at least a/π. 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.