Abstract
We consider growth conditions for (frequently) Birkhoff-universal functions of exponential type with respect to the different rays emanating from the origin. For that purpose, we investigate their (conjugate) indicator diagram or, equivalently, their indicator function. Some known results, where growth is measured with respect to the maximum modulus, are extended.
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References
Ansari, S.I.: Hypercyclic and cyclic vectors. J. Funct. Anal. 128, 374–383 (1995)
Aronszajn, N.: Sur les décompositions des fonctions analytiques uniformes et sur leurs applications. Acta Math. 65, 1–156 (1935)
Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358, 5083–5117 (2006)
Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)
Berenstein, C.A., Gay, R.: Complex analysis and special topics in harmonic analysis. Springer, New York (1995)
Birkhoff, G.D.: Démonstration d’un théorème élémentaire sur les fonctions entières. C.R. Acad. Sci. Paris 189, 473–475 (1929)
Blasco, O., Bonilla, A., Grosse-Erdmann, K.-G.: Rate of growth of frequently hypercyclic functions. Proc. Edinb. Math. Soc. 53, 39–59 (2010)
Boas, R.P.: Entire Functions. Academic Press, New York (1954)
Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergod. Theory Dyn. Syst. 27, 383–404 (2007)
Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors—erratum. Ergod. Theory Dyn. Syst. 29, 1993–1994 (2009)
Chan, K.C., Shapiro, J.H.: The cyclic behaviour of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40, 1421–1449 (1991)
Duyos-Ruiz, S.M.: On the existence of universal functions. Sov. Math. Dokl. 27, 9–13 (1983)
Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos. Springer, London (2011)
Köthe, G.: Dualität in der Funktionentheorie. J. Reine Angew. Math. 191, 30–49 (1953)
Morimoto, M.: An introduction to Sato’s hyperfunctions. Transl. Math. Monogr. American Mathematical Society, Providence (1994)
Müller, J., Naik, S., Ponnusamy, S.: Growth of entire functions of exponential type defined by convolution. Arch. Math. 87, 330–342 (2006)
Müller, J., Wengenroth, J.: Separating singularities of holomorphic functions. Can. Math. Bull. 41, 473–477 (1998)
Rubel, L.A.: Entire and Meromorphic Functions. Springer, New York (1995)
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The author would like to thank the referee for his careful reading of the first version. His suggestions, corrections and his expertise helped improving the presentation significantly.
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Communicated by Stephan Ruscheweyh.
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Beise, HP. Growth of Frequently Birkhoff-Universal Functions of Exponential Type on Rays. Comput. Methods Funct. Theory 13, 21–35 (2013). https://doi.org/10.1007/s40315-012-0001-z
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DOI: https://doi.org/10.1007/s40315-012-0001-z