Sampling Almost Periodic and related Functions

TL;DR

This paper investigates the distribution properties of certain almost periodic functions and constructs sparse sets of real numbers that are useful for sampling theorems, extending to functions generated by complex exponentials including chirps.

Contribution

It establishes conditions for the uniform distribution of finite sets of circle-valued functions and constructs very sparse sets suitable for sampling almost periodic functions and their generalizations.

Findings

Sets can be made arbitrarily sparse, with zero asymptotic density.

Constructed sets are dense in the Bohr group and can be t-sets.

Different almost periodic functions cannot agree on these sets.

Abstract

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, what 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.