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Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent to which it is prevented from exhibiting regions of slow, smooth oscillation around a central average value. One of the issues is that I don't have a very good quantitative definition of what I mean by "flat," which is part of what I'm trying to resolve. Ultimately what I am trying to do is find a good quantitative definition for the "flatness" of a function over some region, and then impose some constraint on this flatness based on the fact that the function is the Fourier transform of a strictly non-negative function.

Long version of question with more details: For reasons involving some physics research I've been working on, I've been dealing with a function $f \left ( t\right )$ which is the Fourier transform of a real, non-negative, and even function,

$f \left ( t \right ) = \int_{-\infty}^{\infty}d \omega ~g \left ( \omega \right )e^{i \omega t}$

That is, the function $g$ has the property of being real, even in $\omega$, and strictly non-negative. I'm aware of some basic results, including the fact that $f$ must be also real, even, etc. I'm also aware of some less trivial results such as Bochner's theorem. But what I'm concerned with is slightly more subtle I think.

Some of the research I've been doing involves situations in which the function $f$ has a very smooth plateau, remaining relatively stationary for a while, with small, slow oscillations around an average value, before later decaying to zero over a relatively short time scale. One test function which I was investigating was the extreme version of this, the rectangle function,

$f_{1} \left ( t \right ) = \left \{ \begin{matrix} 1 & |t| < a \\ 0 & \text{otherwise} \end{matrix} \right.$

In other words, a function which is constant for some window, and zero everywhere else. However, I quickly realized this was not an allowed possibility for the function $f$, because it is the result of a Fourier transform which takes on negative values,

$g_{1} \left ( \omega \right ) = \frac{\sin \left ( a \omega \right )}{\omega }$

(up to some possible factors of $2 \pi$ I admit I'm being lazy about).

What I'm now trying to understand is to what extent I can quantify the notion of "flatness" of a function, and whether or not this is forbidden by being the Fourier transform of a positive function. For example, the function

$f_{1} \left ( t \right ) = \left \{ \begin{matrix} \left (1 + \exp[t - T] \right )^{-1}, & t > 0 \\ \\ \left (1 + \exp[-t - T] \right )^{-1}, & t < 0 \end{matrix} \right.$

has a characteristic "bell shape." Near t = 0, it looks very "flat" for large values of T, having a broad plateau for small times, before later decaying exponentially to zero. For large values of T, the corresponding Fourier transform attains negative values. However, for small values of T, the curve decays more quickly, and intuitively appears much less "flat." There is a crossover value of T at which the Fourier transform appears to never be negative. I've concocted many, many other examples of functions (which I can elaborate on if need be) which seem to associate my intuitive sense of "flatness" with the necessary presence of negative values in the Fourier transform. I've tried to think of various ways to quantify this flatness (such as putting bounds on certain derivatives), but none of the metrics I've thought of seem to be sufficient.

As far as I can tell from fiddling around in Mathematica, $f$ is allowed to decay very quickly, and is also allowed to oscillate very wildly, but staying somewhat stationary is not allowed. More subtle examples involve the fact that $\exp[-t^{2}]$ is allowed, while $\exp[-t^{4}]$, which has a little bit more of a flat "depression" at $t=0$, is not.

Intuitively, I understand what's happening - the Fourier transform of a constant leads to a sinc function, and to the extent that a function "looks flat," its transform will "look like a sinc function," and attain negative values. It would be nice, however, to have some sort of quantitative result that involves these ideas I'm talking about. I've done some reading on auto-correlation functions and characteristic functions (since $g$ could be interpreted as a probability measure), but I haven't yet run across anything like this.

Anyways, sorry for the super long post, and thanks in advance for your help!

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You may check two review papers [1]: Dimitrov and Rusev, The zeros of entire Fourier transforms, EAST JOURNAL ON APPROXIMATIONS Volume 17, Number 1 (2011), 1-108 [2]: Ki, The Zeros of Fourier Transforms. For the definition of flatness, you may try $\delta=\int_{-a}^{a}|f'(z)-f'(0)|$. If f(z)=const, then $\delta=0$ – mike Jun 9 '14 at 16:24

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