The addition formula for theta functions (Q675209)

The author solves the functional equation \(\chi(u+v)\varphi(u-v)= f_1(u)g_1(v)+f_2(u)g_2(v)\), called addition formula for theta functions, under the assumption that \(\chi,\varphi\), \(f_1,f_2,g_1,g_2\) are complex-valued functions on \(\mathbb{R}^n\), \(n\in\mathbb{N}\) arbitrary, and \(\chi\neq 0\) and \(\varphi\neq 0\) are continuous. The main result shows that apart from degeneracy and some obvious modifications, due to the fact that we consider functions on \(\mathbb{R}^n\), theta functions of one complex variable are the only continuous solutions of this functional equation.