Universality and ultradifferentiable functions: Fekete’s theorem

Mouze, A.; Nestoridis, V.

doi:10.1090/S0002-9939-10-10380-3

Universality and ultradifferentiable functions: Fekete’s theorem
HTML articles powered by AMS MathViewer

by A. Mouze and V. Nestoridis PDF

Proc. Amer. Math. Soc. 138 (2010), 3945-3955 Request permission

Abstract:

The purpose of this article is to establish extensions of Fekete’s Theorem concerning the existence of universal power series of $C^{\infty }$ functions defined by estimates on successive derivatives.

References

F. Bayart, K.-G. Grosse-Erdmann, V. Nestoridis, and C. Papadimitropoulos, Abstract theory of universal series and applications, Proc. Lond. Math. Soc. (3) 96 (2008), no. 2, 417–463. MR 2396846, DOI 10.1112/plms/pdm043
P. du Bois Reymond, Über die Fourierschen Reihen, Nachr. Kön. Ges. Wiss. Göttingen 21 (1873) 571-582.
Emile Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. École Norm. Sup. (3) 12 (1895), 9–55 (French). MR 1508908
Charles K. Chui and Milton N. Parnes, Approximation by overconvergence of a power series, J. Math. Anal. Appl. 36 (1971), 693–696. MR 291472, DOI 10.1016/0022-247X(71)90049-7
G. Costakis, M. Marias, and V. Nestoridis, Universal Taylor series on open subsets of $\Bbb R^n$, Analysis (Munich) 26 (2006), no. 3, 401–409. MR 2313217, DOI 10.1524/anly.2006.26.99.401
Karl-Goswin Große-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987), iv+84 (German). Dissertation, University of Trier, Trier, 1987. MR 877464
Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
Jean-Pierre Kahane, Baire’s category theorem and trigonometric series, J. Anal. Math. 80 (2000), 143–182. MR 1771526, DOI 10.1007/BF02791536
Hikosaburo Komatsu, Ultradistributions. I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25–105. MR 320743
Wolfgang Luh, Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten, Mitt. Math. Sem. Giessen 88 (1970), i+56 (German). MR 280692
Vassili Nestoridis, Universal Taylor series, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1293–1306 (English, with English and French summaries). MR 1427126
Vassili Nestoridis and Chris Papadimitropoulos, Abstract theory of universal series and an application to Dirichlet series, C. R. Math. Acad. Sci. Paris 341 (2005), no. 9, 539–543 (English, with English and French summaries). MR 2181390, DOI 10.1016/j.crma.2005.09.028
G. Pál, Zwei kleine Bemerkungen, Tokohu Math. J. 6 (1914/15) 42-43.
Hans-Joachim Petzsche, On E. Borel’s theorem, Math. Ann. 282 (1988), no. 2, 299–313. MR 963018, DOI 10.1007/BF01456977
A. I. Seleznev, On universal power series, Mat. Sbornik N.S. 28(70) (1951), 453–460 (Russian). MR 0041928
N. Tsirivas, Simultaneous approximation by universal series, Math. Nachr., to appear (2010).
Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
A. Zygmund,Trigonometric Series, Cambridge University Press (1979).