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Positive polynomials on Riesz spaces

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Abstract

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem.

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References

  1. Charalambos, D.: Aliprantis and Owen Burkinshaw. Positive operators. Springer, Dordrecht (2006). Reprint of the 1985 original

  2. Benyamini, Y., Lassalle, S., Llavona, J.G.: Homogeneous orthogonally additive polynomials on Banach lattices. Bull. Lond. Math. Soc. 38(3), 459–469 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bochnak, J., Siciak, J.: Polynomials and multilinear mappings in topological vector spaces. Studia Math. 39, 59–76 (1971)

    MathSciNet  MATH  Google Scholar 

  4. Bu, Q., Buskes, G.: Polynomials on Banach lattices and positive tensor products. J. Math. Anal. Appl. 388(2), 845–862 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ercan, Z., Eroğlu, N.: A note on ordered vector spaces and Kantorovich extension theorem. Creat. Math. Inform. 19(1), 33–35 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete mathematics, 2nd edn. Addison-Wesley Publishing Company, Reading (1994)

    MATH  Google Scholar 

  7. Grecu, B.C., Ryan, R.A.: Polynomials on Banach spaces with unconditional bases. Proc. Am. Math. Soc. 133(4), 1083–1091 (2005). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hyers, D.H.: Polynomial operators. In: Topics in mathematical analysis. vol. 11 of Ser. Pure Math. World Sci. Publ., Teaneck, NJ, pp 410–444 (1989)

  9. Kuczma, M.: An introduction to the theory of functional equations and inequalities, 2nd edn. Birkhäuser Verlag, Basel (2009)

    Book  MATH  Google Scholar 

  10. Loane, J.: Polynomials on riesz spaces. PhD thesis, NUI Galway (2008)

  11. Loane, J.: Polynomials on Riesz spaces. J. Math. Anal. Appl. 364(1), 71–78 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mazur, S., Orlicz, W.: Grundlegende eigenschaften der polynomischen operationen. Studia Mathematica 5, 50–68 (1934)

    MATH  Google Scholar 

  13. McKiernan, M.A.: On vanishing \(n{th}\) ordered differences and Hamel bases. Ann. Polon. Math. 19, 331–336 (1967)

    MathSciNet  MATH  Google Scholar 

  14. Riordan J.: Combinatorial identities. Robert E. Krieger Publishing Co., Huntington, NY (1979). Reprint of the 1968 original

  15. Sundaresan, K.: Geometry of spaces of homogeneous polynomials on Banach lattices. In: Applied geometry and discrete mathematics. Vol 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci. Amer. Math. Soc., Providence, RI, pp. 571–586 (1991)

  16. Székelyhidi, L.: Local polynomials and functional equations. Publ. Math. Debrecen, 30(3–4):283–290 (1984) 1983

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

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. National University of Ireland Galway, Galway, Ireland

    James Cruickshank & Raymond A. Ryan

  2. Dundalk Institute of Technology, Dundalk, Ireland

    John Loane

Authors
  1. James Cruickshank
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  2. John Loane
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  3. Raymond A. Ryan
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Correspondence to Raymond A. Ryan.

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Cruickshank, J., Loane, J. & Ryan, R.A. Positive polynomials on Riesz spaces. Positivity 21, 885–895 (2017). https://doi.org/10.1007/s11117-016-0439-8

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  • DOI: https://doi.org/10.1007/s11117-016-0439-8

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