An Integral Formula for the Inhomogeneous Jordan–von Neumann Equation

Abstract

Abstract We study the inhomogeneous form of the Jordan–von Neumann quadratic functional equation, in which the right-hand side is a prescribed function of two real variables. We prove that a twice continuously differentiable solution exists if and only if the prescribed data has the same regularity and satisfies a single three-variable cocycle identity, and we produce a solution in closed form, as an integral expression built from a second-order partial derivative of the data along one coordinate axis. The construction preserves regularity along the standard scale of finitely differentiable, smooth, and polynomial classes. Moreover, the integral formula inverts the quadratic defect operator exactly and selects a canonical solution, namely the unique one whose second derivative vanishes at the origin. As a consequence, the quadratic defect operator restricts to a linear bijection between normalized functions and cocycles in each regularity class. The solution components shared by all solutions are identified, the degree of polynomial solutions is determined exactly, and the reconstruction is shown to depend continuously on the data.