Approximation of a Multivariate Function of Bounded Variation from its Scattered Data

TL;DR

This paper introduces a new scattered data interpolation method capable of approximating any function of bounded variation, extending beyond the native spaces of traditional radial basis function methods.

Contribution

The paper presents a novel interpolation technique that can approximate all functions of bounded variation from scattered data, surpassing the limitations of existing RBF methods.

Findings

The method successfully approximates functions outside RBF native spaces.

It guarantees convergence as data points become dense.

Applicable to a broad class of functions of bounded variation.

Abstract

This paper addresses the problem of approximating a function of bounded variation from its scattered data. Radial basis function(RBF) interpolation methods are known to approximate only functions in their native spaces, and to date, there has been no known proof that they can approximate functions outside the native space associated with the particular RBF being used. In this paper, we describe a scattered data interpolation method which can approximate any function of bounded variation from its scattered data as the data points grow dense. As the class of functions of bounded variation is a much wider class than the native spaces of the RBF, this method provides a crucial advantage over RBF interpolation methods.