# What is the relationship between singularities for complex times and high frequency asymptotics?

As said in a paper I am reading on p 2677 in the text directly above FIG3, this should be a standard result about Fourier transforms of analytic functions. In the paper the authors use these methodes to investigate the intermittent behaviour of turbulent flows kicking in at high frequencies.

But I am not familiar with this, so can somebody explain to me how exactly these singularities in the complex time plane are related to the high frequency behaviour of a function?

• I can't say without reading more context, but this smells like a stationary phase argument: en.wikipedia.org/wiki/Stationary_phase_approximation – Neal Apr 25 '13 at 11:16
• Sorry, the Wikipedia article's not very good. Here are two resources that might be more helpful: math.ku.dk/~gimperlein/dif11/dif11_kim_stationaryphase.pdf and tricki.org/article/… – Neal Apr 25 '13 at 11:22
• You linked to a 33 page article. You can at least tell us on which page you find the passage you don't understand. – Willie Wong Apr 25 '13 at 11:52
• @WillieWong Sorry yes, just a moment ... – Dilaton Apr 25 '13 at 11:55
• Please edit in specific details that you don't get. The basic idea is Paley-Wiener theorem, that the (inverse) Fourier transform of an entire function of growth bounded by exponential should correspond to a function of compact support (and hence decays infinitely fast at high frequencies). Thus one expect the presence of high frequency asymptotics to be tied to the failure of the analytic continuation to be an "entire function of certain growth rate". In particular, singularities would be one way for a function to fail to be entire – Willie Wong Apr 25 '13 at 12:16