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The stability of the sine and cosine functional equations. (English) Zbl 0718.39004

The main result is the following. Let G be an amenable group, \(K={\mathbb{R}}\quad or\quad {\mathbb{C}},\) and let \(f,g: G\to K\) be given functions. The function \((x,y)\to f(xy)-f(x)g(y)-f(y)g(x)\) is bounded if and only if we have one of the following conditions:
(i) \(f=0,g\) is arbitrary;
(ii) \(f,g\) are bounded;
(iii) \(f=am+b,g=m\), where \(a: G\to K\) is additive, \(m: G\rightharpoonup K\) is bounded exponential and b: \(G\to K\) is a bounded function;
(iv) \(f=\lambda m-\lambda b,g=m+b,\) where \(m: G\to K\) is an exponential, \(b: G\to K\) is a bounded function, and \(\lambda \in K;\)
(v) \(f(xy)=f(x)g(y)+f(y)g(x),\) for all x, y in G.
A similar type result is given also for the cosine equation \(f(xy)=f(x)f(y)-g(x)g(y)\).
Reviewer: M.C.Zdun (Kraków)

MSC:

39B32 Functional equations for complex functions
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