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The stability of the sine equation


Author: Piotr W. Cholewa
Journal: Proc. Amer. Math. Soc. 88 (1983), 631-634
MSC: Primary 39B20
DOI: https://doi.org/10.1090/S0002-9939-1983-0702289-8
MathSciNet review: 702289
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Abstract | References | Similar Articles | Additional Information

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

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


References [Enhancements On Off] (What's this?)

  • [1] J. A. Baker, The stability of the sine equation, Proc. Amer. Math. Soc. 80 (1980), 411-416. MR 580995 (81m:39015)
  • [2] J. A. Baker, J. Lawrence and F. Zorzitto, The stability of the equation $ f(x + y) = f(x)f(y)$, Proc. Amer. Math. Soc. 74 (1979), 242-246. MR 524294 (80d:39009)
  • [3] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. MR 0004076 (2:315a)
  • [4] S. M. Ulam, A collection of mathematical problems, Interscience, New York, 1960. MR 0120127 (22:10884)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0702289-8
Keywords: Functional equation, sine equation, stability
Article copyright: © Copyright 1983 American Mathematical Society

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