Convergence of Multilevel Stationary Gaussian Quasi-Interpolation

TL;DR

This paper introduces a new multilevel Gaussian quasi-interpolation method for smooth periodic functions, providing the first error estimates and demonstrating faster convergence than theoretical predictions.

Contribution

It presents the first error analysis for a multilevel Gaussian-based quasi-interpolation algorithm with analytic basis functions.

Findings

Numerical results show faster convergence than theoretical estimates.

The error estimate involves polynomial truncation and remainder control.

The scheme is effective for smooth periodic functions.

Abstract

In this paper we present a new multilevel quasi-interpolation algorithm for smooth periodic functions using scaled Gaussians as basis functions. Recent research in this area has focussed upon implementations using basis function with finite smoothness. In this paper we deliver a first error estimates for the multilevel algorithm using analytic basis functions. The estimate has two parts, one involving the convergence of a low degree polynomial truncation term and one involving the control of the remainder of the truncation as the algorithm proceeds. Thus, numerically one observes a convergent scheme. Numerical results suggest that the scheme converges much faster than the theory shows.