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Extended abstract theory of universal series and applications

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Abstract

In this article, we extend the recently developed abstract theory of universal series to include averaged sums of the form \({\frac{1}{\phi(n)}\sum_{j=0}^{n} a_j x_j}\) for a given fixed sequence of vectors (x j ) in a topological vector space X over a field \({\mathbb{K}}\) of real or complex scalars, where \({(\phi(n))}\) is a sequence of non-zero field scalars. We give necessary and sufficient conditions for the existence of a sequence of coefficients (a j ) which make the above sequence of averaged sums dense in X. When applied, the extended theory gives new analogues to well known classical theorems including those of Seleznev, Fekete and Menchoff.

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References

  1. Armitage D.H.: Universal overconvergence of polynomial expansions of harmonic functions. J. Approx. Theory 118, 225–234 (2002)

    Article  MathSciNet  Google Scholar 

  2. Armitage, D.H.: Dense vector spaces of universal harmonic functions. In: Advances in Multivariate Approximation (Witten-Bommerholz 1998), pp. 33–42 (1999)

  3. Armitage D.H., Gauthier P.M.: Recent developments in harmonic approximation, with applications. Results Math. 29, 1–15 (1996)

    MATH  MathSciNet  Google Scholar 

  4. Bagchi, B.: The Statistical Behaviour and Universality Properties of the Riemann Zeta Function and other Allied Dirichlet Series. Thesis, Indian Statistical Institute, Calcutta (1981)

  5. Bagchi B.: A joint universality theorem for Dirichlet L-functions. Math. Z. 181, 319–334 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bayart F.: Topological and algebraic genericity of divergence and universality. Studia Math. 167, 161–181 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bayart F., Grosse-Erdmann K.-G., Nestoridis V., Papadimitropoulos C.: Abstract theory of universal series and applications. Proc. Lond. Math. Soc. (3) 96, 417–463 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bernal-González L.: Densely hereditarily hypercyclic sequences and large hypercyclic manifolds. Proc. Am. Math. Soc. 127, 3279–3285 (1999)

    Article  MATH  Google Scholar 

  9. Bernal-González L., Calderón-Moreno M.C.: Dense linear manifolds of monsters. J. Approx. Theory 119, 156–180 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bernal-González L., Montes-Rodríguez A.: Non-finite dimensional closed vector spaces of universal functions for composition operators. J. Approx. Theory 82, 375–391 (1995)

    Article  MathSciNet  Google Scholar 

  11. Bourdon P.S.: Invariant manifolds of hypercyclic vectors. Proc. Am. Math. Soc. 118, 845–847 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Čaščina N.S.: On the theory of a universal Dirichlet series (Russian). Izv. Vyssh Uchebn Zaved Mat. 35, 165–167 (1963)

    Google Scholar 

  13. Costakis G., Hadjiloucas D.: Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators. Studia Math. 175, 249–269 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gardiner, S.J.: Harmonic Approximation. LMS Lecture Note Series, vol. 221, CUP (1995)

  15. Grosse-Erdmann K.-G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. (New Series) 36, 345–381 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kahane J.-P.: Baire’s category theorem and trigonometric series. J. d’Analyse Math. 80, 143–182 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Koumoullis G., Luh W., Nestoridis V.: Universal functions are automatically universal in the sense of Menchoff. Complex Variables Elliptic Equ. 52, 307–314 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. León-Saavedra F., Montes-Rodríguez A.: Linear structure of hypercyclic vectors. J. Func. Anal. 148, 524–545 (1997)

    Article  MATH  Google Scholar 

  19. León-Saavedra F., Montes-Rodríguez A.: Spectral theory and hypercyclic subspaces. Trans. Am. Math. Soc. 353, 247–267 (2001)

    Article  MATH  Google Scholar 

  20. Melas A., Nestoridis V.: Universality of Taylor series as a generic property of holomorphic functions. Adv. Math. 157, 138–176 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Menchoff D.: Sur les séries trigonométriques universelles. C. R. (Dokl) Acad. Sci. URSS (NS) 49, 79–82 (1945)

    MathSciNet  Google Scholar 

  22. Menchoff D.: On the convergence in measure of trigonometric series (Russian). Dokl. Akad. Nauk. SSSR (NS) 59, 849–852 (1948)

    Google Scholar 

  23. Menchoff, D.E.: On convergence in measure of trigonometric series (Russian). Trudy Mat. Inst. Steklov. 32 (1950), English transl in: Am. Math. Soc. Transl. (1) 3, 196–270 (1962)

  24. Menchoff, D.: On universal sequences of functions (Russian). Dokl. Akad. Nauk. SSSR 151, 1283–1285 (1963), English transl in: Sov. Math. Dokl. 4, 1191–1193 (1963)

    Google Scholar 

  25. Menchoff, D.E.: Universal sequences of functions (Russian). Mat. Sb (NS) 65(107), 272–312 (1964), English transl in: Am. Math. Soc. Transl. (2) 111, 81–116 (1978)

    Google Scholar 

  26. Montes-Rodríguez A.: Banach spaces of hypercyclic vectors. Michigan Math. J. 43, 419–436 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Montes-Rodríguez, A.: Vector spaces of universal functions. In: Complex Methods in Approximation Theory (Proc Workshop, Almería, 1995), pp. 113–116 (1997)

  28. Nestoridis V.: Universal Taylor series. Ann. Inst. Fourier 46, 1293–1306 (1996)

    MATH  MathSciNet  Google Scholar 

  29. Nestoridis V., Papadimitropoulos C.: Abstract theory of universal series and an application to Dirichlet series. C. R. Math. Acad. Sci. Paris 341, 539–543 (2005)

    MATH  MathSciNet  Google Scholar 

  30. Pál J.: Zwei kleine Bemerkungen. Tôhoku Math. J. 6, 42–43 (1914/15)

    Google Scholar 

  31. Papadimitropoulos, C.: Specific Types of Universal Series in Classical Mathematical Analysis (Greek). Master Thesis, University of Athens (2006)

  32. Seleznev A.I.: On universal power series (Russian). Mat. Sb (NS) 28(70), 453–460 (1951)

    MathSciNet  Google Scholar 

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  1. The School of Sciences, Department of Computer Science and Engineering, European University Cyprus, 6 Diogenes Street, Engomi, P.O. Box 22006, 1516, Nicosia, Cyprus

    Demetris Hadjiloucas

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  1. Demetris Hadjiloucas
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Hadjiloucas, D. Extended abstract theory of universal series and applications. Monatsh Math 158, 151–178 (2009). https://doi.org/10.1007/s00605-008-0038-2

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  • DOI: https://doi.org/10.1007/s00605-008-0038-2

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