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Sampling Almost Periodic and Related Functions

  • Stefano Ferri
  • Jorge GalindoEmail author
  • Camilo Gómez
Article
  • 3 Downloads

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.

Keywords

Almost periodic function Bohr topology Matching set Sampling set Chirp Polynomial phase functions Discrepancy Uniform distribution Sampling T-set Bohr-dense 

Mathematics Subject Classification

Primary 43A60 Secondary 11K70 42A75 94A20 

Notes

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.

References

  1. 1.
    Blum, J.R., Eisenberg, B., Hahn, L.-S.: Ergodic theory and the measure of sets in the Bohr group. Acta Sci. Math. (Szeged) 34, 17–24 (1973)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bohr, H.: Zur Theorie der fastperiodischen Funktionen I. Acta Math. 45, 29–127 (1924)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bohr, H.: Zur Theorie der fastperiodischen Funktionen II. Acta Math. 46, 101–214 (1925)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bohr, H.: Zur Theorie der fastperiodischen Funktionen III. Acta Math. 47, 237–281 (1926)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bohr, H.: Almost Periodic Functions. Chelsea Publishing Company, New York (1947)zbMATHGoogle Scholar
  6. 6.
    Carlen, E., Mendes, R.V.: Signal reconstruction by random sampling in chirp space. Nonlinear Dyn. 56(3), 223–229 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Collet, P.: Sampling almost-periodic functions with random probes of finite density. Proc. R. Soc. Lond. Ser. A 452(1953), 2263–2277 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Drmota, M., Tichy, R.F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Filali, M., Galindo, J.: Approximable \(\mathscr {{WAP}}\)- and \(\mathscr {LUC}\)-interpolation sets. Adv. Math. 233, 87–114 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Graham, C.C., Hare, K.E.: Interpolation and Sidon Sets for Compact Groups. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2013)CrossRefGoogle Scholar
  11. 11.
    Kahane, J.P., Katznelson, Y.: Distribution uniforme de certaines suites d’entiers aléatoires dans le groupe de Bohr. J. Anal. Math. 105, 379–382 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Katznelson, Y.: An Introduction to Harmonic Analysis, corrected edn. Dover Publications Inc., New York (1976)zbMATHGoogle Scholar
  13. 13.
    Katznelson, Y.: Sequences of integers dense in the Bohr group. Proc. R. Inst. Tech. 79–86 (1973)Google Scholar
  14. 14.
    Li, D., Queffélec, H., Rodríguez-Piazza, L.: Some new thin sets of integers in harmonic analysis. J. Anal. Math. 86, 105–138 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    López, J.M., Ross, K.A.: Sidon sets. In: Lecture Notes in Pure and Applied Mathematics, vol. 13. Marcel Dekker Inc., New York (1975)Google Scholar
  16. 16.
    Neuwirth, S.: Two random constructions inside lacunary sets. Ann. Inst. Fourier (Grenoble) 49(6), 1853–1867 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rudin, W.: Weak almost periodic functions and Fourier–Stieltjes transforms. Duke Math. J. 26, 215–220 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rudin, W.: Fourier Analysis on Groups. Wiley, New York. Reprint of the 1962 original, A Wiley-Interscience Publication (1990)Google Scholar
  19. 19.
    Segal, I.E.: The class of functions which are absolutely convergent Fourier transforms. Acta Sci. Math. Szeged 12, no, pp. 157–161. Leopoldo Fejér et Frederico Riesz LXX annos natis dedicatus, Pars B (1950)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de los AndesBogotá D.C.Colombia
  2. 2.Instituto Universitario de Matemáticas y Aplicaciones (IMAC)Universidad Jaume ICastellónSpain
  3. 3.Facultad de IngenieríaUniversidad de La Sabana, Campus Universitario Puente del ComúnChíaColombia
  4. 4.Departamento de MatemáticasEscuela Colombiana de IngenieríaBogotá D.C.Colombia

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