  We introduce intrinsic interpolatory bases for data structured on graphs and
derive properties of those bases. Polyharmonic Lagrange functions are shown to
satisfy exponential decay away from their centers. The decay depends on the
density of the zeros of the Lagrange function, showing that they scale with the
density of the data. These results indicate that Lagrange-type bases are ideal
building blocks for analyzing data on graphs, and we illustrate their use in
kernel-based machine learning applications.
