  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.
