On the approximate solution of D'Alembert type equation originating from number theory (Q2928682)

\textit{L. R. Berrone} and \textit{L. V. Dieulefait} [Aequationes Math. 81, No. 1--2, 167--175 (2011; Zbl 1232.39023)] equipped \(\mathbb{R}^2\) with a composition \(\cdot_\alpha\) for given \(\alpha\in\mathbb{R}\) by NEWLINE\[NEWLINE(x_1,y_1)\cdot_\alpha(x_2,y_2):= (x_1x_2+\alpha y_1y_2,\,x_1y_2+ x_2y_1).NEWLINE\]NEWLINE This makes \((\mathbb{R},\cdot_\alpha)\) into an abelian monoid such that \(\sigma(x,y):= (x,-y)\) is an involution.NEWLINENEWLINE The authors of the present paper find the homomorphisms of \((\mathbb{R}^2,\cdot_\alpha)\) into \((\mathbb{C},\cdot)\). Their form depends on whether \(\alpha>0\), \(\alpha=0\) or \(\alpha<0\). Via \textit{T. M. K. Davison} [Publ. Math. 75, No. 1--2, 41--66 (2009; Zbl 1212.39034)] this enables them to find the solutions \(f:\mathbb{R}^2\to \mathbb{C}\) of the following version of D'Alembert's functional equation: NEWLINE\[NEWLINEE(\alpha): f(a\cdot_\alpha b)+ f(a\cdot_\alpha \sigma(b))= 2f(a) f(b),\quad a,b\in\mathbb{R}^2.NEWLINE\]NEWLINE Finally, they derive the superstability of \(E(\alpha)\).