Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present $\ell_\infty$ error estimates between the local basis and the corresponding interpolatory Lagrange basis functions in cases where the underlying graph satisfies a mild assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate $\|A\|_{\infty} \leq \sqrt{n} \|A\|_{2}$.

翻译：基于图的逼近方法在交通、生物与化学网络、金融模型、图像处理、网络流等多个领域日益受到关注。在这些应用中，逼近空间的基函数通常无法解析获得，必须通过计算得到。我们提出对图上的拉格朗日基函数进行扰动，其中拉格朗日函数来源于一类类似于经典样条的函数。我们考虑的基函数具有局部支撑性质，每个基函数通过求解与图上微分算子相关的小型能量最小化问题获得。在底层图满足关于未知函数顶点连接性的温和假设条件下，我们给出了局部基函数与对应插值型拉格朗日基函数之间的$\ell_\infty$误差估计，并通过数值实验进一步验证了理论边界。我们的分析包含一个正定矩阵的混合范数不等式，该不等式比一般估计$\|A\|_{\infty} \leq \sqrt{n} \|A\|_{2}$更为紧致。