A natural derivative on [0,n] and a binomial Poincaré inequality
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 703-712.

We consider probability measures supported on a finite discrete interval [0, n]. We introduce a new finite difference operator ∇n, defined as a linear combination of left and right finite differences. We show that this operator ∇n plays a key role in a new Poincaré (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.

DOI : 10.1051/ps/2014007
Classification : 46N30, 60E15
Mots clés : discrete measures, transportation, poincaré inequalities, Krawtchouk polynomials 
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     title = {A natural derivative on $[0,~n]$ and a binomial {Poincar\'e} inequality},
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Hillion, Erwan; Johnson, Oliver; Yu, Yaming. A natural derivative on $[0,~n]$ and a binomial Poincaré inequality. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 703-712. doi : 10.1051/ps/2014007. http://www.numdam.org/articles/10.1051/ps/2014007/

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