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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The stability of the exponential equation


Authors: Roman Ger and Peter Semrl
Journal: Proc. Amer. Math. Soc. 124 (1996), 779-787
MSC (1991): Primary 39B72
MathSciNet review: 1291769
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Abstract: We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an $\varepsilon$-approximately exponential function to the set of all exponential functions tends to zero as $\varepsilon$ tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers $\mathbb{Z}$.


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Additional Information

Roman Ger
Affiliation: Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

Peter Semrl
Affiliation: TF, University of Maribor, Smetanova 17, P.O. BOX 224, 62000 Maribor, Slovenia

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03031-6
PII: S 0002-9939(96)03031-6
Keywords: Exponential functions, congruentialy additive functions, stability
Received by editor(s): February 1, 1994
Received by editor(s) in revised form: July 24, 1994
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society