Towards a Fundamental Principle for $λ$-Homogeneous Solutions on Cones
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911724714590208 |
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| author | Tsopanopoulos, Michael |
| author_facet | Tsopanopoulos, Michael |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_28443 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Towards a Fundamental Principle for $λ$-Homogeneous Solutions on Cones Tsopanopoulos, Michael Analysis of PDEs 35E20, 35A08, 35C15, 35A22, 46F12 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. |
| title | Towards a Fundamental Principle for $λ$-Homogeneous Solutions on Cones |
| topic | Analysis of PDEs 35E20, 35A08, 35C15, 35A22, 46F12 |
| url | https://arxiv.org/abs/2605.28443 |