Towards a Fundamental Principle for $\lambda$-Homogeneous Solutions on Cones

Abstract

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