How many Fourier coefficients are needed?

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,…,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0, 1, \ldots , N$$\end{document} are sufficient to determine the function (the intervals). In the second case we produce a set of locations of size O(Nlogd-1N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N \log ^{d-1} N)$$\end{document} which suffices to determine the measure.