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 deter...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2023, Vol.200 (1), p.23-42
Main Authors: Diederichs, Benedikt, Kolountzakis, Mihail N., Papageorgiou, Effie
Format: Article
Language:English
Subjects:
Coefficients
Intervals
Mathematics
Mathematics and Statistics
Toruses
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Summary: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 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.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-022-01792-0