On a C2-nonlinear subdivision scheme avoiding Gibbs oscillations

TL;DR

This paper introduces a new nonlinear subdivision scheme that effectively eliminates Gibbs oscillations near discontinuities while achieving high regularity in its limit functions, with proven convergence and stability.

Contribution

The paper presents the first subdivision scheme that controls Gibbs oscillations and attains a regularity index greater than 1, advancing the handling of discontinuities in approximation schemes.

Findings

Converges to functions with H"older regularity > 1.192

Numerical estimates show regularity index of 2.438

First scheme to control Gibbs phenomenon with high regularity

Abstract

This paper is devoted to the presentation and the analysis of a new nonlinear subdivision scheme eliminating the Gibbs oscillations close to discontinuities. Its convergence, stability and order of approximation are analyzed. It is proved that this schemes converges towards limit functions of H\"older regularity index larger than 1.192. Numerical estimates provide an H\"older regularity index of 2.438. Up to our knowledge, this scheme is the first one that achieves simultaneously the control of the Gibbs phenomenon and regularity index larger than 1 for its limit functions.