  We are looking at families of functions or measures on the torus which are
specified by a finite number of parameters $N$. The task, for a given family,
is to look at a small number of Fourier coefficients of the object, at a set of
locations that is predetermined and may depend only on $N$, and determine the
object. We look at (a) the indicator functions of at most $N$ intervals of the
torus and (b) at sums of at most $N$ complex point masses on the
multidimensional torus. In the first case we reprove a theorem of Courtney
which says that the Fourier coefficients at the locations $0, 1, \ldots, N$ are
sufficient to determine the function (the intervals). In the second case we
produce a set of locations of size $O(N \log^{d-1} N)$ which suffices to
determine the measure.
