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Local polynomials and the Montel theorem

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Abstract

In this paper local polynomials on Abelian groups are characterized by a “local” Fréchet-type functional equation. We apply our result to generalize Montel’s Theorem and to obtain Montel-type theorems on commutative groups.

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References

  1. Almira J.M.: Montel’s theorem and subspaces of distributions which are ∆m-invariant. Numer. Funct. Anal. Optim. 35(4), 389–403 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  2. Almira J.M., Abu-Helaiel K.B.: A note on invariant subspaces and the solution of certain classical functional equations. Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity 11, 3–17 (2013)

    MathSciNet  Google Scholar 

  3. Djokovič, D.Ž.: A representation theorem for (X 1−1)(X 2 −1) · · · (X n −1) and its applications. Ann. Polon. Math. 22, 189–198 (1969/1970)

  4. Fréchet M.: Une definition fonctionelle des polynomes. Nouv. Ann. 9, 145–162 (1909)

    Google Scholar 

  5. Laczkovich M.: Polynomial mappings on abelian groups. Aequationes Math. 68(3), 177–199 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Laczkovich, M.: Local spectral synthesis on Abelian groups. Acta Math. Hung. (2014, to appear)

  7. Mazur S., Orlicz W.: Grundlegende Eigenschaften der polynomischen Operationen I. Studia Math. 5, 50–68 (1934)

    MATH  Google Scholar 

  8. Montel P.: Sur quelques extensions d’un théorème de Jacobi. Prace Matematyczno-Fizyczne 44(1), 315–329 (1935)

    Google Scholar 

  9. Prager W., Schwaiger J.: Generalized polynomials in one and in several variables. Math. Pannon. 20(2), 189–208 (2009)

    MATH  MathSciNet  Google Scholar 

  10. Reich, L., Schwaiger, J.: On polynomials in additive and multiplicative functions. In: Functional Equations: History, Applications and Theory, Math. Appl. Reidel, Dordrecht, pp. 127–160 (1984)

  11. Schwaiger J., Prager W.: Polynomials in additive functions and generalized polynomials. Demonstratio Math. 41(3), 589–613 (2008)

    MATH  MathSciNet  Google Scholar 

  12. Székelyhidi L.: Convolution Type Functional Equations on Topological Abelian Groups. World Scientific Publishing Co. Inc., Teaneck (1991)

    Book  MATH  Google Scholar 

  13. Székelyhidi L.: Polynomial functions and spectral synthesis. Aequationes Math. 70(1–2), 122–130 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Székelyhidi L.: Noetherian rings of polynomial functions on Abelian groups. Aequationes Math. 84(1-2), 41–50 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Vander Lijn G.: La définition fonctionnelle des polynômes dans les groupes abéliens. Fund. Math. 33, 42–50 (1939)

    MathSciNet  Google Scholar 

  16. Waldschmidt, M.: Topologie des Points Rationnels, Cours de Troisième Cycle, Université P. et M. Curie (Paris VI) (1995)

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Jaén, Spain, E.P.S. Linares, C/Alfonso X el Sabio, 28, 23700, Linares (Jaén), Spain

    Jose Maria Almira

  2. Department of Mathematics, University of Debrecen, Debrecen, Hungary

    László Székelyhidi

  3. Department of Mathematics, University of Botswana, P. O. Box 12, Gaborone, Botswana

    László Székelyhidi

Authors
  1. Jose Maria Almira
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  2. László Székelyhidi
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Correspondence to László Székelyhidi.

Additional information

The research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. NK-81402.

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Almira, J.M., Székelyhidi, L. Local polynomials and the Montel theorem. Aequat. Math. 89, 329–338 (2015). https://doi.org/10.1007/s00010-014-0308-0

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  • DOI: https://doi.org/10.1007/s00010-014-0308-0

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