RT Journal Article
SR Electronic
T1 Oscillation of functions with a spectral gap
JF Proceedings of the National Academy of Sciences of the United States of America
JO Proc Natl Acad Sci U S A
FD National Academy of Sciences
SP 5872
OP 5873
DO 10.1073/pnas.0302874101
VO 101
IS 16
A1 Eremenko, A.
A1 Novikov, D.
YR 2004
UL http://www.pnas.org/content/101/16/5872.abstract
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.