High Accuracy Quasi-Interpolation using a new class of generalized Multiquadrics

TL;DR

This paper introduces a new class of generalized multiquadric functions that significantly improve the accuracy and convergence rate of quasi-interpolation in odd-dimensional Euclidean spaces.

Contribution

A novel generalization of multiquadric functions is proposed, enabling polynomial reproduction and enhanced convergence rates for quasi-interpolation.

Findings

Convergence rate improved by a factor of h^{2d-n-1}.

Generalized Fourier transform computed for the new multiquadric.

Detailed analysis conducted on properties using an infinite regular grid.

Abstract

A new generalization of multiquadric functions $\phi(x)=\sqrt{c^{2d}+||x||^{2d}}$, where $x\in\mathbb{R}^n$, $c\in \mathbb{R}$, $d\in \mathbb{N}$, is presented to increase the accuracy of quasi-interpolation further. With the restriction to Euclidean spaces of odd dimensionality, the generalization can be used to generate a quasi-Lagrange operator that reproduces all polynomials of degree $2d-1$. In contrast to the classical multiquadric, the convergence rate of the quasi-interpolation operator can be significantly improved by a factor $h^{2d-n-1}$, where $h>0$ represents the grid spacing. Among other things, we compute the generalized Fourier transform of this new multiquadric function. Finally, an infinite regular grid is employed to analyse the properties of the aforementioned generalization in detail.