D'Alembert's functional equation on restricted domains

Abstract

Summary. In the class of functionals f: X --* ~, where X is an inner product space with dim X > 3, we study the D'Alembert functional equation( 1) on the restricted domainsand x= = {(x, y) ~ xZ/llxll = Ilyll}.In this paper we prove that the equation (1) restricted to X~ is not equivalent to (1) on the whole space 3(. We also succeed in characterizing all common solutions if we add the condition f(2x) = 2f2(x) -1.Using this result, we prove the equivalence between (1) restricted to X 2 and (1) on the whole space X.This research follows similar previous studies concerning the additive, exponential and quadratic functional equations.

Orthogonalities and functional equations

“…It is interesting to compare the families of solutions of the functional equation characterizing the cosine function with its correspondent equation postulated only for orthogonal vectors (see Fochi [72]). …”

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

“…(1) 2 this fact will allow us to use Theorem 3 ( [2]) in order to deduce that h satisfies (1), and therefore (3). Let us now prove that h satisfies (1) 2 .…”

“…It has been proved in [2], by showing examples of functionals satisfying (1) 1 but not (1), that the class A is properly contained in A 1 . The common solutions, i.e.…”

“…The common solutions, i.e. functionals of A 1 belonging to A too, have been characterized in [2] by introducing, as a suitable auxiliary condition, the duplication formula f (2x) = 2f 2 (x) − 1.…”