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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.04342 |
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Table of Contents:
- Let $R$ be a commutative ring with identity and $G$ a graph. \emph{An extending generalized spline} on $G$ is a vertex labeling $f \in \prod_{v} M_v$ such that at each edge $e=uv$ there is an $R$-module $M_{uv}$ together with homomorphisms $ φ_u : M_u \to M_{uv}$ and $ φ_v : M_v \to M_{uv}$ for each vertex $u, v$ incident to the edge $e$ so that $φ_u(f_u)=φ_v(f_v).$ Extending generalized splines are further generalizations for generalized splines. They can also be considered as generalized splines over modules. The main goal of this paper is to study the $R$-module structure of extending generalized splines. We concentrate on two following questions: which of the results for general splines extend to generalized splines over modules and if there is an algorithm or an explicit formula for special basis classes, called a flow up basis, for generalized splines over modules. We show that certain results concerning generalized splines can be extended to a setting where each vertex $v$ is assigned a module $M_v=m_v\mathbb Z$. We provide an algorithm to construct a special basis for generalized splines over these modules on paths. Additionally, we introduce a new technique to construct a flow-up basis on arbitrary graphs using the idea of an algorithm on paths.