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A Functional Equation for the Cosine

Published online by Cambridge University Press:  20 November 2018

PL Kannappan
PL Kannappan*
Affiliation:
University of Waterloo
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It is known [3], [5] that, the complex-valued solutions of

(B)

(apart from the trivial solution f(x)≡0) are of the form

(C)

(D)

In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding ϕ in (C) is also continuous and ϕ is of the form [1],

(E)

In this paper, the functional equation

(P)

where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 3 , August 1968 , pp. 495 - 498
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Aczel, J., Lectures on functional equations and their applications. (Academic Press, 1966).Google Scholar
2. Kaczmarz, S., Sur I'equation fonctionelle f(x) + f(x+y) = ϕ(y)f(x + y/2). Fund. Math., 6 (1924) 122-129.Google Scholar
3. Kannappan, P.L., The functional equation f(xy) + f(xy-1) = 2f(x)f(y) for groups. Proc. Amer. Math. Soc. 19 (1968) 69-74.Google Scholar
4. Van Vleck, B., A functional equation for the sine. Ann. Math. 7 (1910)161-165.Google Scholar
5. Wilson, W.H., On certain related functional equations. Bull. Amer. Math. Soc, 26(1919–1920)300-312.Google Scholar