On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards $C^{s}$ functions with $s>1$

Abstract

Abstract. This paper presents a new nonlinear dyadic subdivision scheme eliminating the Gibbs oscillations close to discontinuities. Its convergence, stability and order of approximation are analyzed. It is proved that this scheme converges towards limit functions Hölder continuous with exponent larger than 1.299. Numerical estimates provide a Hölder exponent of 2.438. This subdivision scheme is the first one that simultaneously achieves the control of the Gibbs phenomenon and has limit functions with Hölder exponent larger than 1.

$(2n-1)$-POINT NONLINEAR TERNARY INTERPOLATING SUBDIVISION SCHEMES

“…It can be replaced either by M GM (x, y) or P P H(x, y). M 2n−1 is a parameter defined in the next section and d (2n−3) f i is representing (2n − 3) th difference for n > 1, for example d (1)…”

“…The convergence for 7-point or 9-point ternary nonlinear interpolating subdivision schemes can be proved following the similar steps. The 5-point nonlinear scheme S N L (3.6) when expressed in the form of equation (2.7), gives us associated 5-point linear interpolating scheme S: 1) and the associated nonlinear function F…”

“…To avoid Gibbs phenomenon while keeping all initial data points in the limiting curve, nonlinear subdivision Schemes like ENO, WENO, PPH and PCHIP ( [1], [2], [3], [4], [5], [6], [8], [9], [10]) were introduced during last several years. Recently we have introduced two families of nonlinear ternary interpolating subdivision schemes to reduce undesirable oscillations.…”

A Family of 5-Point Nonlinear Ternary Interpolating Subdivision Schemes with C2 Smoothness

“…Nonlinear subdivision schemes like ENO, WENO, PPH and PCHIP [1][2][3][4][5][6][7][8][9], were introduced during last several years to address Gibbs phenomenon. The arithmetic mean of second differences was replaced by their harmonic mean in a linear subdivision scheme to change it to a nonlinear scheme in [1,2].…”

“…The arithmetic mean of second differences was replaced by their harmonic mean in a linear subdivision scheme to change it to a nonlinear scheme in [1,2]. The geometric mean of first differences was used instead of the arithmetic mean of first differences in [4].…”