Sampling Almost Periodic and Related Functions
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We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We obtain in this way very sparse sets of real numbers (and of integers) on which two different almost periodic functions cannot agree, which makes them amenable to be used in sampling theorems for these functions. These sets can be made as sparse as to have zero asymptotic density or as to be t-sets, i.e., to be sets that intersect any of their translates in a bounded set. Many of these results are proved not only for almost periodic functions but also for classes of functions generated by more general complex exponential functions, including chirps or polynomial phase functions.
KeywordsAlmost periodic function Bohr topology Matching set Sampling set Chirp Polynomial phase functions Discrepancy Uniform distribution Sampling T-set Bohr-dense
Mathematics Subject ClassificationPrimary 43A60 Secondary 11K70 42A75 94A20
We would like to thank the anonymous referees for the appreciable number of corrections and improvements they suggested. We truly believe their input has resulted in a better paper.
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