What is meaning of a signal having a "finite number of maxima and minima during any single period of time"?

  • $\begingroup$ Is the given solution clear enough? $\endgroup$ – Laurent Duval Jul 16 '17 at 15:00

The Dirichlet conditions provide sufficient conditions under which a real periodic function can be assimilated to its Fourier series at each point of continuity.

One of these conditions is that the functions have a finite number of extrema (maxima or minima) in every bounded interval. This means that, in some sense, the function does not oscillate infinitely on some time segment, even with very tiny oscillations. An example of such an oscillating continuous function can be given by:

$$ f(t)=t^2 \cos (1/t)$$

with $f(0)=0$ by continuity. The ripples around $0$ get narrower and more numerous, and are infinitely many, in any non empty interval containing $0$.

The function is evidently continuous, even differentiable. On each interval defined by $$\left[\frac{1}{(k+1)\pi+\pi/2},\frac{1}{k\pi+\pi/2}\right]$$ the function $f$ vanishes at both ends, and since not identically $0$, it attains at least one extremum on this interval. Hence the function has an infinity of extrema on each interval $[0,a]$, $a>0$. The exact locations of the extremas can be computed with the derivative of $f$.

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