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Journal of Fourier Analysis and Applications

, Volume 23, Issue 1, pp 207–228 | Cite as

The Billard Theorem for Multiple Random Fourier Series

  • Samuel Ronsin
  • Hermine Biermé
  • Lionel MoisanEmail author
Article

Abstract

We propose a generalization of a classical result on random Fourier series, namely the Billard Theorem, for random Fourier series over the d-dimensional torus. We provide an investigation of the independence with respect to a choice of a sequence of partial sums (or method of summation). We also study some probabilistic properties of the resulting sum field such as stationarity and characteristics of the marginal distribution.

Keywords

Billard Theorem Random Fourier series Multiple Fourier series Random phase Random fields 

Mathematics Subject Classification

Primary 42B05 60G60 60G17 Secondary 42B08 60G50 

Notes

Acknowledgments

We thank the anonymous referee for his useful comments, which allowed us to improve the quality of the manuscript. This work was supported by the French National Research Agency grant “MATAIM” ANR-09-BLAN-0029-01.

References

  1. 1.
    Ash, J.M., Welland, G.V.: Convergence, uniqueness, and summability of multiple trigonometric series. Trans. Am. Math. Soc. 163, 401–436 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berens, H., Xu, Y.: Fejèr means for multivariate Fourier series. Math. Z. 221, 449–465 (1996)CrossRefzbMATHGoogle Scholar
  3. 3.
    Billard, P.: Séries de Fourier Aléatoirement Bornées, Continues, Uniformément Convergentes. Studia Math. 22, 310–330 (1963)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Blevins, R.D.: Probability density of finite Fourier series with random phases. J. Sound Vib. 208(4), 617–652 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen, G., Cuny, C.: On Billard’s theorem for Random Fourier series. Bull. Pol. Acad. Sci. Math. 53, 39–53 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fefferman, C.: On the Convergence of Multiple Fourier Series. Bull. Am. Math. Soc. 77(5), 744–745 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fefferman, C.: On the divergence of multiple Fourier series. Bull. Am. Math. Soc. 77(2), 191–195 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Figà-Talamanca, A.: Bounded and continuous random Fourier series on noncommutative groups. Proc. AMS 22, 573–578 (1969)CrossRefzbMATHGoogle Scholar
  9. 9.
    Galerne, B., Gousseau, Y., Morel, J.-M.: Random phase textures: theory and synthesis. IEEE Trans. Image Process. 20(1), 257–267 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Itó, K., Nisio, M.: On the convergence of sums of independent Banach space valued random variables. Osaka J. Math. 5, 35–48 (1968)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kahane, J.-P.: Some Random Series of Functions. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1994)Google Scholar
  12. 12.
    Katznelson, Y.: Sur les Ensembles de Divergence des Séries Trigonométriques. Studia Math. 26, 301–304 (1966)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1991)Google Scholar
  14. 14.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, 2nd edn. AMS, Providence (2001)Google Scholar
  15. 15.
    Lukacs, E.: Characteristic Functions, 2nd edn. Hafner Publishing Co., New York (1970). revised and enlargedzbMATHGoogle Scholar
  16. 16.
    Marcus, M.B., Pisier, G.: Random Fourier Series with Applications to Harmonic Analysis. Annals of Mathematics Studies. Princeton University Press, Princeton (1981)Google Scholar
  17. 17.
    Paley, R., Zygmund, A.: On some series of functions, part I. Proc. Camb. Philos. Soc. 26, 337–357 (1930)CrossRefzbMATHGoogle Scholar
  18. 18.
    Paley, R., Zygmund, A.: On some series of functions, part II. Proc. Camb. Philos. Soc. 26, 458–474 (1930)CrossRefzbMATHGoogle Scholar
  19. 19.
    Paley, R., Zygmund, A.: On some series of functions, part III. Proc. Camb. Philos. Soc. 28, 190–205 (1932)CrossRefzbMATHGoogle Scholar
  20. 20.
    Tevzadze, N.: On the convergence of double Fourier series of quadratic summable functions. Soobsc. Akad. Nauk Gruzin. SSR 77(5), 277–279 (1970)Google Scholar
  21. 21.
    van Wijk, J.: Spot noise texture synthesis for data visualization. SIGGRAPH Comput. Graph. 25(4), 309–318 (1991)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.MAP5, UMR CNRS 8145, Université Paris Descartes, Sorbonne Paris CitéParisFrance
  2. 2.LMA, UMR CNRS 7348, Université de PoitiersChasseneuilFrance

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