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>Canadian Mathematical Bulletin
>Volume 50 Issue 2
>An Infinite Dimensional Vector Space of Universal Functions...
Canadian Mathematical Bulletin
An Infinite Dimensional Vector Space of Universal Functions for H ∞ of the Ball
Published online by Cambridge University Press: 20 November 2018
Richard Aron and
Pamela Gorkin
Richard Aron
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, U.S.A. e-mail: aron@math.kent.edu
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, U.S.A. e-mail: pgorkin@bucknell.edu
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We show that there exists a closed infinite dimensional subspace of ${{H}^{\infty }}\left( {{B}^{n}} \right)$ such that every function of norm one is universal for some sequence of automorphisms of ${{B}^{n}}$ .
47B3847B3346J10
Canadian Mathematical Bulletin , Volume 50 , Issue 2 , 01 June 2007 , pp. 172 - 181
DOI: https://doi.org/10.4153/CMB-2007-018-2[Opens in a new window]
Copyright © Canadian Mathematical Society 2007
[1] Axler, S. and Gorkin, P., Sequences in the maximal ideal space of H∞ . Proc. Amer. Math. Soc. 108(1990), no. 3, 731–740.Google Scholar
[2] Bayart, F., Universal inner functions on the ball. To appear, Canad. Math. Bull.Google Scholar
[3] Bernal-González, L. and Montes-Rodríguez, A., Non-finite-dimensional closed vector spaces of universal functions for composition operators. J. Approx. Theory 82(1995), no. 3, 375–391.CrossRefGoogle Scholar
[4] Birkhoff, G. D., Démonstration d'un théorème élémentaire sur les fonctions entiéres. C. R. Acad. Sci. Paris 189(1929), 473–475.Google Scholar
[5] Chee, P. S., Universal functions in several complex variables. J. Austral. Math. Soc. Ser. A 28(1979), no. 2, 189–196.CrossRefGoogle Scholar
[6] Gauthier, P. M. and Xiao, J., The existence of universal inner functions on the unit ball of Cn . Canad. Math. Bull. 49(2005), no. 3, 409–413.CrossRefGoogle Scholar
[7] Gorkin, P. and Mortini, R. Asymptotic interpolating sequences in uniform algebras. J. London Math. Soc. (2) 67(2003), no. 2, 481–498.CrossRefGoogle Scholar
[8] Gorkin, P. and Mortini, R., Universal Blaschke products. Math. Proc. Cambridge Philos. Soc. 136(2004), no. 1, 175–184.CrossRefGoogle Scholar
[9] Gorkin, P. and Mortini, R., Universal singular inner functions. Canad. Math. Bull. 47(2004), no. 1, 17–21.CrossRefGoogle Scholar
[10] Grosse-Erdmann, K.-G., Universal families and hypercyclic operators. Bull. Amer. Math. Soc. (N.S.) 36(1999), no. 3, 345–381.CrossRefGoogle Scholar
[11] Heins, M., A universal Blaschke product. Archiv Math. 6(1954), 41–44.CrossRefGoogle Scholar
[12] Hosokawa, T., Izuchi, K., and Zheng, D., Isolated points and essential components of composition operators on H∞ . Proc. Amer. Math. Soc. 130(2002), no. 6, 1765–1773 (electronic).CrossRefGoogle Scholar
[13] Montes-Rodríguez, A., Vector spaces of universal functions. In: Complex methods in approximation theory. Monogr. Cienc. Tecnol. 2, Universidad Almería, Almería, 1997, pp. 113–116,.Google Scholar
[14] Montes-Rodríguez, A., A note on Birkhoff open sets. Complex Variables Theory Appl. 30(1996), no. 3, 193–198.CrossRefGoogle Scholar
[15] Mortini, R., Infinite dimensional universal subspaces generated by Blaschke products. Proc. Amer. Math. Soc. 135(2007), no. 6, 1795–1801.CrossRefGoogle Scholar
[16] Rudin, W., Function theory in the unit ball of Cn. Grundlehren der Mathematischen Wissenschaften 241. Springer-Verlag, New York, 1980.CrossRefGoogle Scholar
[17] Seidel, W. and Walsh, J. L., On approximation by euclidean and non-euclidean translations of an analytic function. Bull. Amer. Math. Soc. 47(1941), 916–920.CrossRefGoogle Scholar
Bayart, Frédéric Gorkin, Pamela Grivaux, Sophie and Mortini, Raymond 2009. Bounded universal functions for sequences of holomorphic self-maps of the disk. Arkiv för Matematik, Vol. 47, Issue. 2, p. 205.
Grosse-Erdmann, Karl-G. and Mortini, Raymond 2009. Universal functions for composition operators with non-automorphic symbol. Journal d'Analyse Mathématique, Vol. 107, Issue. 1, p. 355.
Clos, Timothy G. 2020. Universality of automorphisms on the ball of bounded holomorphic functions on the polydisk. Journal of Mathematical Analysis and Applications, Vol. 482, Issue. 2, p. 123581.
Walmsley, David 2020. Large vector spaces generated by bounded universal functions. Journal of Mathematical Analysis and Applications, Vol. 492, Issue. 1, p. 124393.
Chan, Kit C. 2021. Topological Dynamics and Topological Data Analysis. Vol. 350, Issue. , p. 121.
Richard Aron (a1) and Pamela Gorkin (a2)
DOI: https://doi.org/10.4153/CMB-2007-018-2