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Zeros of Normalized Sections of Non Convergent Power Series

  • Alberto DayanEmail author
Article

Abstract

A well known result due to Carlson (C R Acad Sci Paris 178:1677–1680, 1924) affirms that a power series with finite and positive radius of convergence R has no Ostrowski gaps if and only if the sequence of zeros of its nth sections is asymptotically equidistributed to \(\partial \mathbb {D}_R\). Here we extend this characterization to those power series with null radius of convergence, modulo some necessary normalizations of the sequence of the sections of f.

Notes

References

  1. 1.
    Andrievskii, V.V., Blatt, H.P.: Discrepancy of Signed Measures and Polynomial Approximation. Springer, New York (2002)CrossRefGoogle Scholar
  2. 2.
    Bourion, G.: L’ultraconvergence dans les séries de Taylor, Actualités scientifiques et industrielles, vol. 472, Paris (1937)Google Scholar
  3. 3.
    Carlson, F.: Sur quelques suites de polynomes. C. R. Acad. Sci. Paris 178, 1677–1680 (1924)zbMATHGoogle Scholar
  4. 4.
    Dilcher, K., Rubel, L.A.: Zero section of divergent power series. J. Math. Anal. Appl. 198(1), 98–110 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Erdös, P., Fried, H.: On the connection between gaps in power series and the roots of their partial sums. Trans. Am. Math. Soc. 62(1), 53–61 (1947)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fernández, J.L.: Zeros of sections of power series: deterministic and random. Comput. Methods Funct. Theory 17(3), 463–486 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jentzsch, R.: Untersuchungen zur Theorie der Folgen analytischer Funktionen. Acta Math. 41, 219 (1918)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Marden, M.: Geometry of Polynomials. Mathematical Surveys and Monographs, vol. 3. American Mathematical Society, Providence (1966)zbMATHGoogle Scholar
  9. 9.
    Robbins, H.: A remark on Stirling’s Formula. Am. Math. Mon. 62(1), 26–29 (1955)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Szegö, G.: Über die Nullstellen von Polynomen, die in einem Kreis gleichmässig konvergieren. Sitzungsber. Berliner Math. Ges. 21, 59–64 (1922)zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsWashington University in St. LouisSt. LouisUSA

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