An abstract approach to Bohr's phenomenon

Authors:
L. Aizenberg, A. Aytuna and P. Djakov

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2611-2619

MSC (2000):
Primary 32A37, 32A05; Secondary 46E10

Published electronically:
March 1, 2000

MathSciNet review:
1657738

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Abstract | References | Similar Articles | Additional Information

Abstract: In 1914 Bohr discovered that there exists such that if a power series converges in the unit disk and its sum has modulus less than , then for the sum of absolute values of its terms is again less than . Recently analogous results were obtained for functions of several variables. Our aim here is to present an abstract approach to the problem and show that Bohr's phenomenon occurs under very general conditions.

**1.**L. Aizenberg, Multidimensional analogues of Bohr's theorem on power series, Proc. Amer. Math. Soc., to appear. CMP**98:16****2.**L. Aizenberg, A. Aytuna, P. Djakov, Generalization of Bohr's theorem for arbitrary bases in spaces of holomorphic functions of several variables, preprint METU, Mathematics 98/168.**3.**Harold P. Boas and Dmitry Khavinson,*Bohr’s power series theorem in several variables*, Proc. Amer. Math. Soc.**125**(1997), no. 10, 2975–2979. MR**1443371**, 10.1090/S0002-9939-97-04270-6**4.**H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2)**13**(1914) 1-5.**5.**C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene werte nicht annemen, Math. Ann.**64**(1907), 95-115.**6.**Peter L. Duren,*Univalent functions*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR**708494****7.**Reinhold Meise and Dietmar Vogt,*Introduction to functional analysis*, Oxford Graduate Texts in Mathematics, vol. 2, The Clarendon Press, Oxford University Press, New York, 1997. Translated from the German by M. S. Ramanujan and revised by the authors. MR**1483073****8.**E. C. Titchmarsh, The theory of functions, Oxford University Press, 1939.

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Additional Information

**L. Aizenberg**

Affiliation:
Department of Mathematics and Computer Science, Bar-Ilan University, 52900Ramat-Gan, Israel

Email:
aizenbrg@macs.biu.ac.il

**A. Aytuna**

Affiliation:
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Email:
aytuna@rorqual.cc.metu.edu.tr

**P. Djakov**

Affiliation:
Department of Mathematics, Sofia University, 1164 Sofia, Bulgaria

Email:
djakov@fmi.uni-sofia.bg

DOI:
https://doi.org/10.1090/S0002-9939-00-05270-9

Keywords:
Spaces of holomorphic functions,
Bohr phenomenon

Received by editor(s):
July 17, 1998

Received by editor(s) in revised form:
October 15, 1998

Published electronically:
March 1, 2000

Additional Notes:
The first author’s research was supported by the BSF grant No 94-00113

The second author wishes to thank L. Aizenberg and institute E. Noether for the invitation to Bar-Ilan University and their hospitality during his visit in Israel

The third author’s research was supported in part by NRF of Bulgaria, grant no. MM-808/98

Communicated by:
Steven R. Bell

Article copyright:
© Copyright 2000
American Mathematical Society