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

Summary.

Taking into account some recent results concerning the conditional d'Alembert equation on orthogonal vectors ¶¶\( f(x + y) + f(x - y) = 2 f(x) f(y) \) for all \( x,y \in X \) with (x,y) = 0 (1)₁¶in the class of the functionals f from a real inner product space X into \( {\Bbb R} \), we deal here with two other conditional functional equations related to (1)₁, namely ¶¶\( g : X \to {\Bbb R}, \quad g(x + y) g(x - y) = g^2(x) + g^2(y) \) for \( x,y \in X \) with (x,y) = 0(2)₁¶¶ and¶¶\( h : X \to {\Bbb R}, \quad h(x + y) h(x - y) = h^2(x) + h^2(y) - 1 \) for \( x,y \in X \) with (x,y) = 0,(3)₁¶where we denote by \( g^2(\cdot) \) and \( h^2(\cdot) \) the pointwise product of functions.¶Our main purpose in this note is to find and characterize some special classes of solutions of the above equations (2)₁ and (3)₁.¶We remark that some results are obtained in the class of the functionals defined on a real inner product space X with \( \dim X \ge 2 \), while other theorems are proved if \( \dim X \ge 3 \).