New methods for quasi-interpolation approximations: resolution of odd-degree singularities

TL;DR

This paper introduces new quasi-interpolation methods using radial basis functions to address odd-degree singularities, improving approximation accuracy and polynomial precision.

Contribution

It develops explicit Fourier-based constructions for quasi-Lagrange functions to handle singularities not matching the parity of the space dimension.

Findings

New quasi-Lagrange functions for singularities

Enhanced approximation quality and polynomial precision

Insights into algebraic and analytic properties of expansions

Abstract

In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange functions when the orders of the singularities of the radial function's Fourier transforms at zero do not match the parity of the dimension of the space, and therefore new expansions and coefficients are needed to overcome this problem. We develop explicit constructions of infinite Fourier expansions that provide these coefficients and make an extensive comparison of the approximation qualities and - with a particular focus - polynomial precision and uniform approximation order of the various formulae. One of the interesting observations concerns the link between algebraic conditions of expansion coefficients and analytic properties of localness and convergence.