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 $e^{⟨x,z⟩}$ with homogeneous kernels $(-⟨x,z⟩{)}^{\lambda }$. 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.