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Suppose $f(t)$ is a function whose Fourier transform $\hat{f}(\omega) = \int_{-\infty}^{+\infty} f(t) e^{- \omega t} dt$ is supported on the interval $[-\epsilon,+\epsilon]$. Is there a theorem to the effect that $f$ can't change too fast? Intuitively, it seems like this ought to be the case since $f$ is ``composed'' only out of slowly varying sines and cosines.

For example, will it be true that if we restrict $||f||=1$ for some appropriate norm, then some bound on $\sup_t |f'(t)|$ will hold? If not, are there other ways to make the preceeding paragraph precise?

P.S. This is related to a different question I asked a while ago.

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@robinson: I'm not sure I understand your precise question. Here's my two cents anyways. If $f$ is in the Schwartz space, then so is $\widehat{f}$. Also, if $f$ is compactly supported, then $\widehat{f}$ is NOT compactly supported. All this really tells us is that since $\widehat{f}$ is compactly supported, then $f$ is not compactly supported, but it is a Schwartz function (assuming $\widehat{f}$ is smooth). I'm not sure if that helps any or not. In your example, what is $\| f \|$? Is it the $L^1$ norm? $L^2$? – JavaMan Feb 13 '12 at 22:02
@JavaMan: I believe that he means to say that if $\hat{f}$ is compactly supported, can one prove any sort of bound on $|f'|$. He is willing to allow the assumption that $||f||$ is bounded in some sort of norm. – Eric Haengel Feb 13 '12 at 22:09
@Dilip Sarwate - Thanks! If you post your comment as an answer, I would love to accept it. – robinson Feb 14 '12 at 0:41
up vote 1 down vote accepted

A theorem due to Bernstein says that if a bounded function $x(t)$ has Fourier transform $$X(f) = \int_{-\infty}^{\infty} x(t)\exp(-i 2\pi ft) \mathrm dt$$ with bounded suppport: $X(f) = 0$ for $|f| > W$, then $$\max \left |\frac{\mathrm dx(t)}{\mathrm dt} \right | \leq 2\pi W \max |x(t)|.$$ This result is stated in Temes, Barcilon, and Marshall, The optimization of bandlimited systems, Proc. Inst. Electrical and Electronics Engineers, Feb. 1973. I don't have the paper anymore and cannot tell you which publication of Bernstein (or older paper or textbook) is cited by Temes, Barcilon and Marshall as the source of the result.

Note that the maximum magnitude of the derivative of $A \sin(2\pi Wt)$ is $2\pi W |A|$.

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