How many Fourier coefficients are needed?

TL;DR

This paper investigates the minimal number of Fourier coefficients needed to uniquely determine certain families of functions or measures on the torus, providing explicit bounds and reproofs of existing theorems.

Contribution

It establishes bounds on the number of Fourier coefficients required for unique determination of indicator functions and measures on the torus, including a new set of locations for measures.

Findings

Fourier coefficients at 0 to N determine indicator functions of N intervals.

A set of size O(N log^{d-1} N) suffices to determine measures of N point masses.

Reproof of Courtney's theorem on Fourier coefficient sufficiency.

Abstract

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.