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I'm recently encountering such a problem, which I think is very intuitive but I have no background knowledge on this field:

Given a signal with certain frequency distribution, e.g. we know that the signal has only high frequency part, we can expect that the number of zero-crossings of the signal has to be large. Is there a way to quantify this statement? For example, Can we give a (deterministic or stochastic) lower bound on the zero-crossings?

Thanks a lot for any hint or reference.

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In the discrete frequency setting (periodic signal) everything is nice and clean: since we can find a real-valued trigonometric polynomial $P$ of degree $\le n$ with given $2n$ or fewer roots on the circle, the real-valued signal $f=\sum_k a_k z^k$ that changes sign at most than $2n$ times, should have some non-zero $a_k$ with $|k|\le n$ (just take the polynomial $P$ with the same sign changes and observe that $fP$ preserves sign, so $\int fP\ne 0$) and that is sharp ($z^{n+1}+z^{-(n+1)}$ has $2n+2$ zeroes and the number of sign changes is always even).

The continuous case is much messier. What exactly do you need?

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  • $\begingroup$ Thanks Fedja. Discrete result is enough for me. I was actually expecting some condition on the distribution on Fourier coefficients of the function, e.g. if we know \sum ka_k^2 is large instead of a_k=0 for |k|<=N, can we say something on the zero crossings, but the current statement is useful as well. Thanks again. $\endgroup$
    – David
    Commented Jan 17, 2011 at 0:46

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