An integral formula for the inhomogeneous Jordan--von Neumann equation

Abstract

We study the inhomogeneous form of the Jordan--von Neumann quadratic functional equation, in which the right-hand side is a prescribed function $g$ of two real variables. We prove that the existence of a $C^{2}$ solution is equivalent to $g$ being itself of class $C^{2}$ and satisfying a single three-variable cocycle identity, and we exhibit the solution as a closed-form integral expression involving the second partial derivative of $g $ along the first coordinate axis. The construction preserves regularity along the standard scale of $C^{k}$, smooth, and polynomial classes.