A natural derivative on [0, Â n] and a binomial PoincarÃ© inequality. (22nd October 2014) - Hillion, Erwan
Abstract We consider probability measures supported on a finite discrete interval [0, Â n]. We introduce a new finite difference operator âˆ‡n, defined as a linear combination of left and right finite differences. We show that this operator âˆ‡n plays a key role in a new PoincarÃ© (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures.