$C^{\infty}$ rational approximation and quasi-histopolation of functions with jumps through multinode Shepard functions

TL;DR

This paper introduces a smooth rational quasi-histopolation method using multinode Shepard functions to effectively approximate functions with jumps, reducing oscillations and overcoming classical approximation phenomena.

Contribution

The paper presents a novel $C^{ abla}$$ ext{ }$rational quasi-histopolation operator that blends local histopolation polynomials with Shepard functions to improve approximation of discontinuous functions.

Findings

Reduces Runge and Gibbs phenomena in function approximation.

Demonstrates high accuracy through numerical experiments.

Provides a flexible approach for bounded function reconstruction.

Abstract

Histopolation, or interpolation on segments, is a mathematical technique used to approximate a function $f$ over a given interval $I=[a,b]$ by exploiting integral information over a set of subintervals of $I$. Unlike classical polynomial interpolation, which is based on pointwise function evaluations, histopolation reconstructs a function using integral data. However, similar to classical polynomial interpolation, histopolation suffers from the well-known Runge phenomenon when integral data are based on a grid with many equispaced nodes, as well as the Gibbs phenomenon when approximating discontinuous functions. In contrast, quasi-histopolation is designed to relax the strict requirement of passing through all the given data points. This inherent flexibility can reduce the likelihood of oscillatory behavior using, for example, rational approximation operators. In this work, we introduce a $C^{\infty}$ rational quasi-histopolation operator, for bounded (integrable) functions, which reconstruct a function by defeating both the Runge and Gibbs phenomena. A key element of our approach is to blend local histopolation polynomials on a few nodes using multinode Shepard functions as blending functions. Several numerical experiments demonstrate the accuracy of our method.