  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.
