How many Fourier coefficients are needed?

Abstract

A. 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, . . . , 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.