A note on d'Alembert's functional equation on a restricted domain (Q744041)

The article deals with the functional equation \[ f(x + y) + f(x - y) = 2f(x)g(y), \qquad (x,y) \in G \times A, \] where \(f:\;G \to P\), \(g:\;A \to P\), \(G\) is a commutative group, \(A\) its subgroup, \(P\) a quadratically closed field. The main result is the following: If \(\text{char}\, P \neq 2\), then one of the following two statements is valid: (i) there are an exponential \(\mu:\;A \to \{-1,1\}\), a family of additive functions \(L_\sigma:\;A \to P\) for \(\sigma \in G/A\), a lifting \(\xi:\;G/A \to G\) with \(\xi([0]) = 0\), and a function \(\alpha:\;G/A \to P\) with \(f(w) = \mu(w - \xi([w]))(L_{|w|}(w - \xi([w])) - \alpha([w])\) for \(w \in G\) amd \(g = \mu\); (b) there exist an exponential \(m:\;A \to P \setminus \{0\}\), a lifting \(\xi:\;G / A \to G\) with \(\xi([0]) = 0\), and functions \(\beta,\gamma:\;G/A \to P\) such that \(g = m_e\) and \(f(w) = \beta([w])m_o(w - \xi([w])) + \gamma([w])m_a(w - \xi([w]))\) for \(w \in G\) (\(m_e(x) = \frac12 \, (m(x) + m(-x))\), \(m_o(x) = \frac12 \, (m(x) - m(-x))\)). The case \(\text{char}\, P = 2\) is also condsidered.