On solutions of the second generalization of d'Alembert's functional equation on a restricted domain

Let A be a subgroup of an abelian group (G, +) and P be a quadratically closed field with char P not equal 2. We give a full description of all pairs of functions f : G -> P, g : A -> P satisfying the equation f(x + y) + f(x - y) = 2g(x)f(y) (x,y) is an element of A x G. We present an example of solution (f, g) of (a) that cannot be extended to a solution (f, (g) over bar) of the equation f(x + y) + f(x - y) = 2 (g) over bar (x)f(y) x,y is an element of G. (C) 2013 Elsevier Inc. All rights reserved.