Abstract
We unify the recently developed abstract theories of universal series and extended universal series to include sums of the form \(\sum _{k=0}^n a_k x_{n,k}\) for given sequences of vectors \((x_{n,k})_{n\ge k\ge 0}\) in a topological vector space \(X\). The algebraic and topological genericity as well as the spaceability are discussed. Then we provide various examples of such generalized universal series which do not proceed from the classical theory. In particular, we build universal series involving Bernstein’s polynomials, we obtain a universal series version of MacLane’s Theorem, and we extend a result of Tsirivas concerning universal Taylor series on simply connected domains, exploiting Bernstein-Walsh quantitative approximation theorem.
Similar content being viewed by others
References
Bayart, F., Grosse-Erdmann, K.-G., Nestoridis, V., Papadimitropoulos, C.: Abstract theory of universal series and applications. Proc. Lond. Math. Soc. 96, 417–463 (2008)
Bayart, F., Matheron, É.: Matheron, dynamics of linear operators, cambridge tracts in math. Cambridge University Press, Cambridge (2009)
Charpentier, S.: On the closed subspaces of universal series in Banach and Fréchet spaces. Studia. Math. 198, 121–145 (2010)
Charpentier, S., Menet, Q., Mouze, A.: Closed universal subspaces of spaces of infinitely differentiable functions. Ann. Inst. Fourier. 64(no1), 297–325 (2014)
Costakis, G., Hadjiloucas, D.: Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators. Studia. Math. 175, 249–269 (2006)
Grosse Erdmann, K.G.: Grosse erdmann, holomorphe monster und Universelle Funktionen. Mitt. Math. Sem. Giersen. 176, 1–84 (1987)
Grosse Erdmann, K.G.: Universal families and hypercyclic operator. Bull. Amer. Math. Soc. 3, 345–381 (1999)
Hadjiloucas, D.: Extended abstract theory of universal series and applications. Monat. Math. 158(2), 151–178 (2009)
Kahane, J.-P.: Baire’s category theorem and trigonometric series. J. Anal. Math. 80, 143–182 (2000)
Korovkin, P.P.: Linear operators and approximation theory. Hindustan Publishing Corporation, Delhi (1960)
Léon-Saavedra, F.: Operators with hypercyclic Cesaro means. Studia. Math. 152, 201–215 (2002)
Luh, W.: Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten. Mitt. Math. sem. Giessen. 88, 1–56 (1970)
Luh, W.: Universal approximation properties of overconvergent power series on open sets. Analysis. 6, 191–207 (1986)
MacLane, G.R.: Sequences of derivatives and normal families. J. Analyse. Math. 2, 72–87 (1952)
Melas, A., Nestoridis, V.: Universality of Taylor Series as a generic property of holomorphic functions. Adv. Math. 157, 138–176 (2001)
Menet, Q.: Sous-espaces fermés de séries universelles sur un espace de Fréchet. Studia. Math. 207, 181–195 (2011)
Menet, Q.: Hypercyclic subspaces and weighted shifts. Adv. Math. 255, 305–337 (2014)
Mouze, A.: Polynomial approximations and universality. Studia. Math. 196, 103–120 (2010)
Müller, J., Yavrian, A.: On polynomials sequences with restricted growth near infinity. Bull. Lond. Math. Soc. 34, 189–199 (2002)
Nestoridis, V.: Universal Taylor series. Ann. Inst. Fourier. 46(5), 1293–1306 (1996)
Nestoridis, V.: An extension of the notion of universal Taylor series, Computational methods and function theory. Ser. Approx. Decompos. 11, 421–430 (1997)
Pál, G.: Zwei kleine Bemerkungen. Tohoku. Math. J. 6, 42–43 (1914)
Ransford, T.: Ransford, potential theory in the complex plane. Cambridge University Press, Cambridge (1995)
Tsirivas, N.: A generalization of universal Taylor series in simply connected domains. J. Math. Anal. Appl. 388, 361–369 (2012)
Walsh, J.L.: Interpolation and approximation by rational functions in the complex domain. American Mathematical Society, New York (1960)
Acknowledgments
The authors gratefully acknowledge the referee for her/his careful reading of the manuscript and her/his many valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Teschl
Rights and permissions
About this article
Cite this article
Charpentier, S., Mouze, A. & Munnier, V. Generalized universal series. Monatsh Math 179, 15–40 (2016). https://doi.org/10.1007/s00605-015-0764-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-015-0764-1