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