Sums of trigonometric functions may or may not be periodic functions; in particular, $\sin(ax)+\sin(bx)$ is periodic if $a/b$ is rational.

If we consider the function \begin{equation} f(x) = \sin(3x) + \sin(\pi x) \end{equation} it surely looks periodic, even if it's not; to me it feels like the period itself is somewhat periodic (or is the result of a kind of "period cascade").

My question is: is there some way to capture this "quasiperiodic" nature of this kind of functions, i.e. does there exist a measure of how "repetitive" a function is even if it is not a periodic function?

To narrow the scope of the question, I'm trying at the moment to figure out the case of the above sum of sines.

  • $\begingroup$ How about: a function is pseudoperiodic if it is the sum of finitely many periodic functions? $\endgroup$ – Mees de Vries Jan 11 '17 at 16:09
  • $\begingroup$ Just an observation in the case of your specific example. The longer of the two periods, between the places where the sum almost vanishes, is close to the value of $\dfrac{2\pi}{\pi-3}$. I tried one other, $\sin(2x)+\sin\left(\frac{\pi}{2}\right)$ and the longer period is close to $\dfrac{2\pi}{2-\frac{\pi}{2}}$. Perhaps it would be worthwhile to check out this relationship $\dfrac{2\pi}{\vert b-a\vert}$ with other values of $a$ and $b$. $\endgroup$ – John Wayland Bales Jan 11 '17 at 16:39
  • $\begingroup$ However, I see a 'period' of $\frac{1}{4}\cdot\dfrac{2\pi}{\frac{\pi}{3}-1}$ for $\sin\left(\dfrac{\pi}{3} x\right)+\sin(x)$. $\endgroup$ – John Wayland Bales Jan 11 '17 at 16:51
  • $\begingroup$ The point is that one can come up with a local approximation of a "period" but this approximation will vanish on bigger scales. $\endgroup$ – marco trevi Jan 11 '17 at 16:59
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    $\begingroup$ In the Bohr-Weyl development of almost-periodic functions, the criterion is $|f(t+T)-f(t)|< \epsilon.$ I wonder if this is related to your question? $\endgroup$ – daniel Jan 13 '17 at 10:16

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