Comonotone and coconvex rational interpolation and approximation

Abstract

Comonotonicity and coconvexity are well-understood in uniform polynomial approximation and in piecewise interpolation. The covariance of a global (Hermite) rational interpolant under certain transformations, such as taking the reciprocal, is well-known, but its comonotonicity and its coconvexity are much less studied. In this paper we show how the barycentric weights in global rational (interval) interpolation can be chosen so as to guarantee the absence of unwanted poles and at the same time deliver comonotone and/or coconvex interpolants. In addition the rational (interval) interpolant is wellsuited to reflect asymptotic behaviour or the like.

Determining and benchmarking risk neutral distributions implied from option prices

“…Unfortunately, the theoretical number of discrete conditions needed for this implication to hold is often too high to be of practical use. Guaranteeing a rational approximation with a nonnegativity second derivative on the entire real line is currently out of scope here, but the interested reader is referred to [37]. For fixed ' and m, the problem that remains, is to obtain nonzero values for the coefficients of r ';m ðKÞ such that the homogeneous linear inequalities (7), (9) and (11) are satisfied.…”

Positive interpolation by trigonometric functions

“…Manni [10] proposed a parametric C 2 Hermite interpolation technique for shape preservation of monotone and convex 2D data. Nguyen et al [12] studied the co-monotonicity and co-convexity for a global (Hermite) rational interpolant. The authors in [12] assured the absence of unwanted poles and the preservation of co-monotonicity and/or co-convexity through suitable selection of weights in global rational interpolation.…”

“…Nguyen et al [12] studied the co-monotonicity and co-convexity for a global (Hermite) rational interpolant. The authors in [12] assured the absence of unwanted poles and the preservation of co-monotonicity and/or co-convexity through suitable selection of weights in global rational interpolation. Zhu et al [18] constructed quartic trigonometric polynomial blending functions analogous to the ordinary quintic Bernstein polynomial basis functions.…”

Visualization of constrained data by smooth rational fractal interpolation

“…In the context of shape preserving and constrained control for curves and surfaces, the rational functions are more effective than the polynomial functions. Many literatures on the rational interpolation have investigated the problem [1,2,10,11,14,15,[19][20][21]. And yet, it is not well explored hitherto in references on the fractal interpolation.…”