Advertisement

Zeros of Normalized Sections of Non Convergent Power Series

  • 15 Accesses

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.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Notes

  1. 1.

    We may choose any \(1<r<\infty \) if G is null.

References

  1. 1.

    Andrievskii, V.V., Blatt, H.P.: Discrepancy of Signed Measures and Polynomial Approximation. Springer, New York (2002)

  2. 2.

    Bourion, G.: L’ultraconvergence dans les séries de Taylor, Actualités scientifiques et industrielles, vol. 472, Paris (1937)

  3. 3.

    Carlson, F.: Sur quelques suites de polynomes. C. R. Acad. Sci. Paris 178, 1677–1680 (1924)

  4. 4.

    Dilcher, K., Rubel, L.A.: Zero section of divergent power series. J. Math. Anal. Appl. 198(1), 98–110 (1996)

  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)

  6. 6.

    Fernández, J.L.: Zeros of sections of power series: deterministic and random. Comput. Methods Funct. Theory 17(3), 463–486 (2017)

  7. 7.

    Jentzsch, R.: Untersuchungen zur Theorie der Folgen analytischer Funktionen. Acta Math. 41, 219 (1918)

  8. 8.

    Marden, M.: Geometry of Polynomials. Mathematical Surveys and Monographs, vol. 3. American Mathematical Society, Providence (1966)

  9. 9.

    Robbins, H.: A remark on Stirling’s Formula. Am. Math. Mon. 62(1), 26–29 (1955)

  10. 10.

    Szegö, G.: Über die Nullstellen von Polynomen, die in einem Kreis gleichmässig konvergieren. Sitzungsber. Berliner Math. Ges. 21, 59–64 (1922)

Download references

Author information

Correspondence to Alberto Dayan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research supported by Starting Grant (StG), PE1, ERC-2012-StG-20111012.

Communicated by Irene Sabadini, Michael Shapiro and Daniele Struppa.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dayan, A. Zeros of Normalized Sections of Non Convergent Power Series. Complex Anal. Oper. Theory 14, 10 (2020) doi:10.1007/s11785-019-00963-6

Download citation