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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 sine equation


Author: Piotr W. Cholewa
Journal: Proc. Amer. Math. Soc. 88 (1983), 631-634
MSC: Primary 39B20
MathSciNet review: 702289
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Abstract: Let $ \delta $ be a positive real constant and let $ G$ be an abelian group (written additively) in which division by 2 is uniquely performable. Every unbounded complex-valued function $ f$ on $ G$ satisfying the inequality

$\displaystyle \left\vert {f(x + y)f(x - y) - f{{(x)}^2} + f{{(y)}^2}} \right\vert \leqslant \delta \quad {\text{for all }}x,y \in G$

has to be a solution of the sine functional equation

$\displaystyle f(x + y)f(x - y) = f{(x)^2} - f{(y)^2}\quad {\text{for all }}x,y \in G.$


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0702289-8
PII: S 0002-9939(1983)0702289-8
Keywords: Functional equation, sine equation, stability
Article copyright: © Copyright 1983 American Mathematical Society