A $C^{\infty}$ rational quasi-interpolation operator for functions with jumps without the Gibbs phenomenon

TL;DR

This paper introduces a smooth rational quasi-interpolation operator that effectively approximates functions with jumps, avoiding the Gibbs phenomenon and improving accuracy in handling discontinuities in numerical analysis.

Contribution

The paper proposes a novel $C^{ abla}$ rational quasi-interpolation operator specifically designed for functions with jumps, enhancing approximation quality without Gibbs phenomenon artifacts.

Findings

Successfully approximates functions with jumps

Reduces Gibbs phenomenon effects

Provides smooth and accurate interpolations

Abstract

The study of quasi-interpolation has gained significant importance in numerical analysis and approximation theory due to its versatile applications in scientific and engineering fields. This technique provides a flexible and efficient alternative to traditional interpolation methods by approximating data points without requiring the approximated function to pass exactly through them. This approach is particularly valuable for handling jump discontinuities, where classical interpolation methods often fail due to the Gibbs phenomenon. These discontinuities are common in practical scenarios such as signal processing and computational physics. In this paper, we present a $C^{\infty}$ rational quasi-interpolation operator designed to effectively approximate functions with jump discontinuities while minimizing the issues typically associated with traditional interpolation methods.