Univariate interpolation by exponential functions and gaussian RBFs for generic sets of nodes

TL;DR

This paper develops a theoretical framework for univariate interpolation using Gaussian RBFs and exponential functions on arbitrary nodes, providing error formulas and convergence proofs, with applications to optimization.

Contribution

It introduces closed-form error expressions and proves exponential convergence for analytic functions, extending the understanding of RBF and exponential interpolation.

Findings

Derived explicit interpolation error formulas.

Proved exponential convergence for analytic functions.

Applied results to optimization algorithms.

Abstract

We consider interpolation of univariate functions on arbitrary sets of nodes by Gaussian radial basis functions or by exponential functions. We derive closed-form expressions for the interpolation error based on the Harish-Chandra-Itzykson-Zuber formula. We then prove the exponential convergence of interpolation for functions analytic in a sufficiently large domain. As an application, we prove the global exponential convergence of optimization by expected improvement for such functions.