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

Technical Summary: An Integral Formula for the Inhomogeneous Jordan–von Neumann Equation

Problem Statement
This paper investigates the inhomogeneous version of the classical Jordan–von Neumann quadratic functional equation. The homogeneous equation, $f(x+y) + f(x-y) = 2f(x) + 2f(y)$, characterizes norms arising from inner products and admits solutions of the form $f(x) = B(x,x)$ for a symmetric bilinear form $B$. The authors consider the inhomogeneous problem where a prescribed function $g: \mathbb{R}^2 \to \mathbb{R}$ acts as the right-hand side:
$$f(x+y) + f(x-y) - 2f(x) - 2f(y) = g(x,y).$$
The central questions are: under what conditions does a solution $f$ exist for a given $g$, and can such a solution be explicitly constructed? The problem is framed within the context of functional equation stability (perturbation of the homogeneous equation) and parallels the classical inverse problem for the linear Cauchy operator $L[f](x,y) = f(x+y) - f(x) - f(y)$.

Methodology and Theoretical Framework
The authors define a linear operator $Q$ such that $Q[f] = g$. The kernel of $Q$ corresponds to the solution space of the homogeneous equation. The methodology relies on:

1. Cocycle Identities: Drawing an analogy to the linear case where the image of the Cauchy operator is characterized by the 2-cocycle identity (related to associativity), the authors derive a three-variable "cocycle identity" specific to the quadratic operator $Q$.
2. Regularity Assumptions: The analysis is conducted within the class of twice continuously differentiable functions ($C^2$).
3. Differentiation and Integration: The core technique involves differentiating the derived cocycle identity with respect to one variable at zero to reduce the functional equation to a differential relation, followed by integration to recover the solution $f$.

Key Contributions and Results

• Derivation of the Quadratic Cocycle Identity: The paper establishes that if $g$ lies in the image of $Q$ (i.e., $g = Q[f]$), it must satisfy the following identity for all $x, y, z \in \mathbb{R}$:
  $$g(x+y,z) + g(x-y,z) + 2g(x,y) = g(x,y+z) + g(x,y-z) + 2g(y,z).$$
  This identity is shown to be the necessary condition for $g$ to be in the image of $Q$.

• Main Theorem (Existence and Construction): The primary result (Theorem 1.1) asserts that for a function $g \in C^2(\mathbb{R}^2)$, the cocycle identity is both necessary and sufficient for the existence of a solution $f \in C^2(\mathbb{R})$.

  • Necessity: Proven by expanding both sides of the identity using the definition of $Q[f]$.
  • Sufficiency and Integral Formula: If $g \in C^2(\mathbb{R}^2)$ satisfies the identity, a solution is explicitly given by the closed-form integral:
    $$f(x) = -\frac{1}{2}g(0,0) - \frac{1}{2}(\partial_2 g)(0,0)x + \frac{1}{2} \int_0^x (x-t)(\partial_2^2 g)(t,0) \, dt.$$
  • Uniqueness: The solution is unique up to the addition of an element from $\ker Q$ (i.e., a homogeneous solution).

• Regularity Preservation: The construction preserves regularity. If $g$ belongs to $C^k(\mathbb{R}^2)$ ($k \ge 2$), $C^\infty(\mathbb{R}^2)$, or is a real polynomial, the resulting $f$ belongs to the corresponding class ($C^k(\mathbb{R})$, $C^\infty(\mathbb{R})$, or polynomials, respectively).

• Operatorial Interpretation: The paper provides a unifying operatorial view, interpreting the cocycle identity as the symmetry of iterated defects of the operator $T$ along coordinate axes. This generalizes the standard 2-cocycle condition for the linear operator $L$ to the quadratic operator $Q$.

Significance and Limitations
The paper claims to provide the first explicit integral representation for the inhomogeneous Jordan–von Neumann equation under $C^2$ regularity, analogous to results by Prunescu for the linear Cauchy equation. The significance lies in converting a functional equation problem into a constructive calculus problem involving partial derivatives and integrals.

The authors explicitly note the following boundaries and open questions:

• Regularity Constraints: The current integral formula requires $g \in C^2$. The case for $g \in C^1$ remains open, as the derivation relies on second-order differentiation.
• Algebraic vs. Regularity Conditions: It remains an open question whether the cocycle identity alone (without any regularity hypothesis on $g$) is sufficient for $g$ to be in the image of $Q$. The authors note that for the linear case, regularity is required to enforce symmetry; a similar obstruction involving Hamel bases may exist for the quadratic case.
• Continuous Case: Whether the cocycle identity implies symmetry (and thus solvability) for merely continuous $g$ is an attractive open problem, mirroring Erdős's work on the linear case.

The paper concludes by verifying the theorem with specific examples (polynomial, exponential, and trigonometric $g$) and outlining these directions for future research.