Equally spaced points are optimal for Brownian Bridge kernel interpolation

In this paper we show how ideas from spline theory can be used to construct a local basis for the space of translates of a general iterated Brownian Bridge kernel $k_{\beta,\varepsilon}$ for $\beta\in\mathbb{N}$, $\varepsilon\geq 0$. In the simple case $\beta=1$, we derive an explicit formula for the corresponding Lagrange basis, which allows us to solve interpolation problems without inverting any linear system. We use this basis to prove that interpolation with $k_{1,\varepsilon}$ is uniformly stable, i.e., the Lebesgue constant is bounded independently of the number an location of the interpolation points, and that equally spaced points are the unique minimizers of the associated power function, and are thus error optimal. In this derivation, we investigate the role of the shape parameter $\varepsilon>0$, and discuss its effect on these error and stability bounds. Some of the ideas discussed in this paper could be extended to more general Green kernels.