Stability of a generalized quadratic functional equation in Schwartz distributions
The author investigates the Hyers-Ulam stability of the generalized quadratic functional equation \[ f(x + y) + g(x - y) - 2h(x) - 2k(y) = 0 \qquad (x,y\in\mathbb{R}^n) \] in the Schwartz distributional sense. The main result states that the distributions \(u,v,w,k\in \mathcal{D}'(\mathbb{R}^n)\) satisfy \[ \|u\circ A + v\circ B - 2w\circ P_1 - 2\kappa\circ P_2\|_{L^\infty}\leq\varepsilon, \] (where \(\circ\) denotes the pullback and \(A,B,P_1,P_2\) are defined by \(A(x,y):=x+y\), \(B(x,y):=x-y\), \(P_1(x,y):=x\), and \(P_2(x,y):=y\)) if and only if there exist a quadratic function \(Q:\mathbb{R}^n\to \mathbb{C}\), constants \(a,b\in\mathbb{C}^n\), \(c_1,c_2,c_3,c_4\in\mathbb{C}\) and measurable functions \(r_1,r_2,r_3,r_4:\mathbb{R}^n\to \mathbb{C}\) such that \[ u=Q(x)+(a+b)\cdot x+c_1+r_1(x),\qquad v=Q(x)+(a-b)\cdot x+c_2+r_2(x), \] \[ w=Q(x)+a\cdot x+c_3+r_3(x),\qquad \kappa=Q(x)+b\cdot x+c_4+r_4(x), \] \[ \|r_1\|_{L^\infty}\leq8\varepsilon,\qquad \|r_2\|_{L^\infty}\leq8\varepsilon,\qquad \|r_3\|_{L^\infty}\leq4\varepsilon,\qquad \|r_4\|_{L^\infty}\leq4\varepsilon, \] and \[ \|r_1(x+y)+r_2(x-y)-2r_3(x)-2r_4(y)+c_1+c_2-2c_3-2c_4\|_{L^\infty} \leq8\varepsilon. \]
generalized quadratic functional equation
scientific article; zbMATH DE number 5645771
scientific article