A Multidimensional Functional Equation Having Quadratic Forms as Solutions

Abstract

We obtain the general solution and the stability of the m-variable quadratic functional equation f(x1+y1,…,xm+ym)+f(x1−y1,…,xm−ym)=2f(x1,…,xm)+2f(y1,…,ym). The quadratic form f(x1,…,xm)=∑1≤i≤j≤maijxixj is a solution of the given functional equation.