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Stability of a generalized trigonometric functional equation. (English) Zbl 1192.39025

The authors establish the stability of the equation \[ \mathcal{F}(x+y)-\mathcal{G}(x-y)=2\mathcal{H}(x)\mathcal{K}(y), \] where \( \mathcal{F}, \mathcal{G}, \mathcal{H}, \mathcal{K}\) are nonzero functions from an abelian group \(G\) to \(\mathbb{C}\). In fact the equation above contains the known equations such as the sine functional equation, the D’Alembert functional equation and the Wilson functional equation. In the last section they investigate some applications of their main results to the framework of Banach algebras.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
39B32 Functional equations for complex functions
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