Locally supported, quasi-interpolatory bases for the approximation of functions on graphs

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 an 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 ${\Vert A\Vert }_{\infty }\le \sqrt{n}{\Vert A\Vert }_2$.