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

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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?

This post imported from StackExchange Mathematics at 2014-12-13 01:36 (UTC), posted by SE-user Dilaton

asked Apr 25, 2013 in Mathematics by Dilaton (6,240 points) [ no revision ]

I can't say without reading more context, but this smells like a stationary phase argument: en.wikipedia.org/wiki/Stationary_phase_approximation

This post imported from StackExchange Mathematics at 2014-12-13 01:37 (UTC), posted by SE-user Neal

commented Apr 25, 2013 by Neal (0 points) [ no revision ]

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/…

This post imported from StackExchange Mathematics at 2014-12-13 01:37 (UTC), posted by SE-user Neal

commented Apr 25, 2013 by Neal (0 points) [ no revision ]

You linked to a 33 page article. You can at least tell us on which page you find the passage you don't understand.

This post imported from StackExchange Mathematics at 2014-12-13 01:37 (UTC), posted by SE-user Willie Wong

commented Apr 25, 2013 by Willie Wong (580 points) [ no revision ]

@WillieWong Sorry yes, just a moment ...

This post imported from StackExchange Mathematics at 2014-12-13 01:37 (UTC), posted by SE-user Dilaton

commented Apr 25, 2013 by Dilaton (6,240 points) [ no revision ]

Also, the article you linked to gave a quick discussion of the relevant concepts in Section IIc, and refers to References 35 and 36 for background. Have you consulted them first? If so, what are your specific questions? As you can see, reference 36 is a book; unless you pinpoint where you don't understand I'm afraid this question would be way too broad to be answered in the Q&A format of this forum.

This post imported from StackExchange Mathematics at 2014-12-13 01:37 (UTC), posted by SE-user Willie Wong

commented Apr 25, 2013 by Willie Wong (580 points) [ no revision ]

@WillieWong I am just rereading the paper after getting an overview to anderstand the details now. I'd rather like to avoid having to read a whole book to understand this article. I thouht this is a relatively easy thing I just dont get. Sure, I can edit in more specific details about what I dont get when I'm at Section IIc again for example. Or should I delete the post ... :-/?

This post imported from StackExchange Mathematics at 2014-12-13 01:37 (UTC), posted by SE-user Dilaton

commented Apr 25, 2013 by Dilaton (6,240 points) [ no revision ]

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

This post imported from StackExchange Mathematics at 2014-12-13 01:37 (UTC), posted by SE-user Willie Wong

commented Apr 25, 2013 by Willie Wong (580 points) [ no revision ]

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