We are looking at families of functions or measures on the torus (in dimension one and two) 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 two-dimensional 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 $O(N \log N)$ which suffices to determine the measure.

翻译：我们研究的是(一维和二维的)横线上的功能或措施的大小,这些功能或措施由一定数目的参数确定,其数量以美元计算。对于一个特定的家庭来说,任务是在一套预先确定并可能仅依赖美元的地点,在一组地点查看该物体的少量四倍系数,并确定该物体。我们研究的是(a) 横线中最多以美元为间隔的指标功能,以及(b) 在两维横线上以最多以美元复合点质量计算。在第一个案例中,我们重新验证了Courtney的一个理论,该理论说,四倍系数在这些地点是0,1,\ldots,N$足以确定该函数(间隔)。在第二个案例中,我们生产出一套大小为$O(N)的大小位置,足以确定尺度。