Volume 18, 2014
|Page(s)||703 - 712|
|Published online||22 October 2014|
A natural derivative on [0, n] and a binomial Poincaré inequality
University of Luxembourg, Campus Kirchberg 1359, Luxembourg
2 Statistics Group, Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK
3 Department of Statistics, University of California, Irvine, CA 92697, USA
Revised: 5 January 2014
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.
Mathematics Subject Classification: 46N30 / 60E15
Key words: Discrete measures / transportation / Poincaré inequalities / Krawtchouk polynomials
© EDP Sciences, SMAI 2014
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