1
$\begingroup$

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.

  • 0
    @Dilip Sarwate - Thanks! If you post your comment as an answer, I would love to accept it.2012-02-14

1 Answers 1

1

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 $2\pi W |A|$.