** Up:** Variable stars: Which Nyquist

The key function is the spectral window:

| |
(2) |

We have and Replacing with , we notice that
this function is periodic with period 1/*p*
(each individual term in the sum has this same periodicity).
Moreover as *g*(*t*) is real, the function
is symmetric around the origin.
These properties of periodicity and symmetry imply that
is also symmetric around (2*k*+1)/2*p*. For *k*=0,
we have 1/2*p*, which is just the Nyquist frequency.
For the power spectrum, the proof is exactly the same.
We can think of the set *t*_{i} as a regular sampling with
step *p*, where some (most) data have been dropped. The more
random the data is, the smaller *p* will be and so the higher
.As in the regular case, the continuous signal *f*(*t*) should not
contain any component above . Otherwise, it will be
mirrored into , and, without
further assumptions,
it will not be distinguishable from the mirror image. Of course,
no frequency component filtered during the measurement process
can be recovered by the random sampling. In particular, if
is the exposure time of every
measures, then in the frequency
domain, the signal is filtered by and
not much information will be recovered above
.

** Up:** Variable stars: Which Nyquist

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