On two functional equations originating from number theory (Q2018807)

If \(f:\mathbb{R}^2 \to \mathbb{R}\) satisfies the functional equation \[ f(x_1, y_1) f(x_2, y_2) = f(x_1 x_2 + y_1 y_2, x_1 y_2 - x_2 y_1)\tag{1} \] for \(x_1, y_1, x_2, y_2 \in \mathbb{R}\), then there exists a multiplicative function \(m: \mathbb{R} \to \mathbb{R}\) such that \[ f(x,y) = m(\sqrt{x^2 + y^2}), \;x,y \in \mathbb{R}. \] Let \(\phi: \mathbb{R}^2 \to \mathbb{R}^+\). If \(f:\mathbb R^2 \to \mathbb{R}\) is an unbounded function satisfying one of the inequalities \[ |f(x_1, y_1) f(x_2, y_2) - f(x_1 x_2 + y_1 y_2, x_1 y_2 - x_2 y_1)| \leq \phi(x_1, y_1), \;x_1, x_2, y_1, y_2 \in \mathbb{R}, \] \[ \;|f(x_1, y_1) f(x_2, y_2) - f(x_1 x_2 + y_1 y_2, x_1 y_2 - x_2 y_1)| \leq \phi(x_2, y_2), \;\;x_1, x_2, y_1, y_2 \in \mathbb{R}, \] then \(f\) is of the form (1). Similar results are obtained for real functions of four variables.