Extending Generalized Splines Over The Integers

TL;DR

This paper introduces extending generalized splines over modules on graphs, explores their module structure, and provides algorithms for constructing special bases, extending known results to new module settings.

Contribution

It generalizes the concept of splines over modules, extends existing results, and develops algorithms for basis construction on paths and arbitrary graphs.

Findings

Results extend to splines over modules with vertex modules as multiples of integers.

An algorithm for constructing a basis on path graphs is provided.

A new technique for basis construction on arbitrary graphs is introduced.

Abstract

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 $ \varphi_u : M_u \to M_{uv}$ and $ \varphi_v : M_v \to M_{uv}$ for each vertex $u, v$ incident to the edge $e$ so that $\varphi_u(f_u)=\varphi_v(f_v).$ 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.