Equally spaced points are optimal for Brownian Bridge kernel interpolation

TL;DR

This paper demonstrates that equally spaced points are optimal for interpolation using Brownian Bridge kernels, providing explicit formulas and stability analysis, and showing their error minimization properties.

Contribution

It introduces a local basis for Brownian Bridge kernels, derives explicit Lagrange basis formulas, and proves the optimality of equally spaced points for interpolation stability and error minimization.

Findings

Equally spaced points minimize the interpolation error for Brownian Bridge kernels.

The interpolation with $k_{1, ext{epsilon}}$ is uniformly stable regardless of point configuration.

Explicit Lagrange basis formulas enable error-free interpolation without solving linear systems.

Abstract

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.