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I am using the forecast package in R to get some Fourier components - namely, function fourier(ts, K, ..). For a time series ts, K controls the maximum number of the Fourier terms one gets (sines + cosines of with different frequencies).

However, if the period of the time series is specified as 7, K is automatically capped at 3 maximum, or for other periods, $\#\textrm{terms} = \frac{\textrm{Period}}{2}$.

Why is that? I thought Fourier series can be represented by an infinite number of components. Is there some practical issue?

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You are observing a discrete series. With a period of $k$ and a constant term already fitted, there are only $k-1$ degrees of freedom for the seasonal terms (there's only $k$ different means you can have in a period with $k$ times, but one is accounted for by the constant). This is the same df you get if you used seasonal dummies.

Because you have both $\sin$ and $\cos$ terms, you normally get two terms each harmonic. So with $k=7$, you get $(k-1)/2=3\,$ $\sin$ terms and $(k-1)/2=3\,$ $\cos$ terms. (With even $k$, you get one term at the end rather than two, because one - the highest order $\sin$ term - doesn't contribute anything.)

Any higher order periodicities would consist of wiggles between the data points. You don't observe those.

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  • $\begingroup$ Ah, so it is due to the function being discrete? So, as I understood, the first 7 terms approximate all 7 points of the period, while higher order terms try to approximate something which is in-between points, so it makes no sense (since the ts is discrete), and they are simply discarded. Did I get it correctly? $\endgroup$ – SWIM S. Jan 20 at 3:03
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    $\begingroup$ Essentially, yes; being on a discrete set of points you can only determine as many coefficients as points - and because the points are on a lattice, higher harmonics couldn't do anything anyway. $\endgroup$ – Glen_b Jan 20 at 5:14

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