Fabry gap theorem

In analysis, the Fabry gap theorem is a result about the analytic continuation of lacunary power series. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on its circle of convergence. It was proved by Eugène Fabry in 1896.

Statement of the theorem

Let 0<n0<n1<… be a sequence of integers such that the sequence nk/k diverges to ∞, or equivalentlyD:=lim supk→∞knk=0.Let (ak)k≥0 be a sequence of complex numbers such that the power series

f(z)=∑k=0∞akznk

has radius of convergence 1. Then the unit circle is a natural boundary for the series f.

The theorem may be deduced from the first main theorem of Turán's method.

Converse statement

An exact converse to the theorem was established by George Pólya in 1942. If D>0, then there exists a power series with exponent sequence (nk)k≥0 of radius of convergence equal to 1, but for which the unit circle is not a natural boundary.

Pólya's gap theorem

Pólya's gap theorem, also known as the Fabry–Pólya theorem, is a refinement of the Fabry gap theorem. The theorem was proven by Fabry in 1899 and put in its modern form by Pólya in 1929, who also supplied a simpler proof. It states that every arc of the circle of convergence of length greater than 2πD contains a singularity of f. When D=0, it follows that the singularities are dense on the circle of convergence, which is the conclusion of the Fabry gap theorem. This extension also bears a converse.

See also

References

  1. ↑ Fabry, Eugène (1896). "Sur les points singuliers d'une fonction donnée par son développement en série et l'impossibilité du prolongement analytique dans des cas très généraux". Annales scientifiques de l'École normale supérieure. 13: 367–399. doi:10.24033/asens.431. ISSN 0012-9593.
  2. ↑ Erdős, P. (1945). "Note on the Converse of Fabry's Gap Theorem". Transactions of the American Mathematical Society. 57 (1): 102. doi:10.2307/1990169.
  3. ↑ Boas, Ralph P. (1954). Entire functions. New York: Academic Press. ISBN 978-0-12-108150-8. {{cite book}}: ISBN / Date incompatibility (help)
  4. ↑ Pólya, George (1942). "On converse gap theorems". Transactions of the American Mathematical Society. 52 (1): 65–71. doi:10.1090/S0002-9947-1942-0006577-0. ISSN 1088-6850.
  5. 1 2 Koosis, Paul (1992). The Logarithmic Integral II. Cambridge, New York: Cambridge University Press. ISBN 978-0-521-30907-3.
  6. 1 2 Martirosian, Valeri A. (2005-05-01). "The converse of the Fabry-Pólya theorem on singularities of lacunary power series". Analysis. 25 (2): 123–130. doi:10.1515/anly-2005-0203. ISSN 2196-6753.
  7. ↑ Fabry, E. (1899). "Sur les séries de Taylor qui ont une infinité de points singuliers". Acta Mathematica. 22: 65–87. doi:10.1007/BF02417871. ISSN 0001-5962.
  8. ↑ Pólya, G. (1929). "Untersuchungen über Lücken und Singularitäten von Potenzreihen". Mathematische Zeitschrift (in German). 29 (1): 549–640. doi:10.1007/BF01180553. ISSN 0025-5874.