Characterization of special classes of solutions for some functional equations on orthogonal vectors (Q1972867)

The author deals with two functional equations restricted to the orthogonal vectors, namely \[ g(x+y)g(x-y)=\bigl(g(x) \bigr)^2+ \bigl(g(y) \bigr)^2 \] (for all \(x,y\in X\) with \((x,y)=0)\) and \[ g(x+y)g(x-y)=\bigl( g(x) \bigr)^2 +\bigl(g(y) \bigr)^2-1, \] (for all \(x,y\in X\) with \((x,y)=0)\), where \(g:X\to \mathbb{R}\) and \(X\) is a real inner product space. Some special classes of the above equations are characterized. This is a continuation of the author's previous research about the d'Alembert equation \[ g(x+y)+ g(x-y)= 2g(x)g(y) \] (for all \(x,y\in X\) with \((x,y)=0)\).