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Sampling Almost Periodic and Related Functions

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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, 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.

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Acknowledgements

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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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de los Andes, Cra 1 18A-10, Apartado Aéreo 4976, Bogotá D.C., Colombia

    Stefano Ferri & Camilo Gómez

  2. Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, 12071, Castellón, Spain

    Jorge Galindo

  3. Facultad de Ingeniería, Universidad de La Sabana, Campus Universitario Puente del Común, Chía, Colombia

    Camilo Gómez

  4. Departamento de Matemáticas, Escuela Colombiana de Ingeniería, AK 45 205-59, Bogotá D.C., Colombia

    Camilo Gómez

Authors
  1. Stefano Ferri
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  2. Jorge Galindo
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  3. Camilo Gómez
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Correspondence to Jorge Galindo.

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Communicated by Stephane Jaffard.

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Research of Stefano Ferri was supported by the Faculty of Sciences of Universidad de los Andes via the Proyecto Semilla: “Representabilidad de grupos topológicos y de Álgebras de Banach y aplicaciones.” Research of Jorge Galindo was author supported by Ministerio de Economía y Competitividad (Spain) through Project MTM2016-77143-P (AEI/FEDER, UE).

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Ferri, S., Galindo, J. & Gómez, C. Sampling Almost Periodic and Related Functions. Constr Approx 52, 213–232 (2020). https://doi.org/10.1007/s00365-019-09479-w

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  • DOI: https://doi.org/10.1007/s00365-019-09479-w

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