Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The stability of the cosine equation


Author: John A. Baker
Journal: Proc. Amer. Math. Soc. 80 (1980), 411-416
MSC: Primary 39B70; Secondary 39B20
DOI: https://doi.org/10.1090/S0002-9939-1980-0580995-3
MathSciNet review: 580995
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $ \delta > 0$, G is an abelian group and f is a complex-valued function defined on G such that $ \vert f(x + y) + f(x - y) - 2f(x)f(y)\vert \leqslant \delta $ for all $ x,y \in G$, then either $ \vert f(x)\vert \leqslant (1 + \sqrt {1 + 2\delta } )/2$ for all $ x \in G$ or $ f(x + y) + f(x - y) = 2f(x)f(y)$ for all $ x,y \in G$.


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

  • [1] John 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)
  • [2] 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)
  • [3] Pl. Kannappan, The functional equation $ f(xy) + f(x{y^{ - 1}}) = 2f(x)f(y)$ for groups, Proc. Amer. Math. Soc. 19 (1968), 69-74. MR 0219936 (36:3006)
  • [4] S. M. Ulam, A collection of mathematical problems, Interscience, New York, 1960. MR 0120127 (22:10884)
  • [5] W. H. Wilson, On certain related functional equations, Bull. Amer. Math. Soc. 26 (1919), 300-312. MR 1560309

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 39B70, 39B20

Retrieve articles in all journals with MSC: 39B70, 39B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0580995-3
Keywords: Functional equation, cosine equation, stability
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society