Journal d'Analyse Mathématique

, Volume 133, Issue 1, pp 27–49 | Cite as

Generalised continuation by means of right limits

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Abstract

Several theories have been proposed to generalise the concept of analytic continuation to holomorphic functions of the disc for which the circle is a natural boundary. Elaborating on Breuer-Simon’s work on right limits of power series, Baladi-Marmi-Sauzin recently introduced the notion of renascent right limit and rrl-continuation. We discuss a few examples and consider particularly the classical example of Poincaré simple pole series in this light. These functions are represented in the disc as series of infinitely many simple poles located on the circle; they appear, for instance, in small divisor problems in dynamics. We prove that any such function admits a unique rrl-continuation, which coincides with the function obtained outside the disc by summing the simple pole expansion. We also discuss the relation with monogenic regularity in the sense of Borel.

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References

  1. [Ag49]
    S. Agmon, Sur les séries de Dirichlet, Ann. Sci. École Norm. Sup. (3), 66 (1949), 263–310.MathSciNetCrossRefMATHGoogle Scholar
  2. [BMS12]
    V. Baladi, S. Marmi, and D. Sauzin, Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps, Ergodic Theory and Dynam. Systems, 34 (2014), 777–800.MathSciNetCrossRefMATHGoogle Scholar
  3. [Bo17]
    E. Borel, Leçons sur les fonctions monogènes uniformes d’une variable complexe, Gauthier-Villars, Paris, 1917.MATHGoogle Scholar
  4. [BS11]
    J. Breuer and B. Simon, Natural boundaries and spectral theory, Adv. Math., 226 (2011), 4902–4920.MathSciNetCrossRefMATHGoogle Scholar
  5. [CMS14]
    C. Carminati, S. Marmi, and D. Sauzin, There is only one KAM curve, Nonlinearity, 27 (2014), 2035–2062.MathSciNetCrossRefMATHGoogle Scholar
  6. [HP57]
    E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, American Mathematical Society, Providence, RI, 1957.MATHGoogle Scholar
  7. [MS03]
    S. Marmi and D. Sauzin, Quasianalytic monogenic solutions of a cohomological equation, Mem. Amer. Math. Soc., 164 (2003), no. 780.Google Scholar
  8. [MS11]
    S. Marmi and D. Sauzin, A quasianalyticity property for monogenic solutions of small divisor problems, Bull. Braz. Math. Soc. (N.S.) 42 (2011), 45–74.MathSciNetCrossRefMATHGoogle Scholar
  9. [MT88]
    J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems (College Park, MD, 1986–87), Springer, Berlin, 1988, pp. 465–563.CrossRefGoogle Scholar
  10. [Po83]
    H. Poincaré, Sur les fonctions à espaces lacunaires, Acta Soc. Fenn., 12 (1883), 343–350.Google Scholar
  11. [Po92]
    H. Poincaré, Sur les fonctions à espaces lacunaires, Amer. J. Math., 14 (1892), 201–221.MathSciNetCrossRefMATHGoogle Scholar
  12. [RS02]
    W. T. Ross and H. S. Shapiro, Generalized Analytic Continuation, American Mathematical Society, Providence, RI, 2002.MATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2017

Authors and Affiliations

  1. 1.CNRS IMCCE, Observatoire de ParisParisFrance
  2. 2.University of TorontoTorontoCanada

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