Towards a Fundamental Principle for $λ$-Homogeneous Solutions on Cones

Abstract

We prove a weak fundamental principle for $λ$-homogeneous solutions of homogeneous constant-coefficient systems on open pointed convex cones. Starting with the solution family $S_{\mathcal B}$ arising in the Ehrenpreis--Palamodov theory, we construct a corresponding family $S_{\mathcal B,λ}$ by replacing the exponential kernels $e^{\langle x,z\rangle}$ with homogeneous kernels $(-\langle x,z\rangle)^λ$. The key tool is a Mellin-type operator on Paley--Wiener spaces, which links the classical theory to the Euler-constrained setting. For $λ\in \mathbb{C}\setminus \mathbb{N}_0$ and under a visibility assumption, we show that the span of $S_{\mathcal B,λ}$ is dense in the space of $λ$-homogeneous solutions.