A natural derivative on [0,n] and a binomial Poincar\'e inequality

TL;DR

This paper introduces a new finite-difference operator on [0,n], establishing a binomial Poincaré inequality with Krawtchouk polynomials as eigenfunctions, and explores its connection to optimal transport.

Contribution

It presents a novel finite-difference operator and a corresponding Poincaré inequality for binomial measures, linking spectral analysis and optimal transport.

Findings

The operator $ abla_n$ is key to a new spectral gap inequality.

Krawtchouk polynomials are eigenfunctions of the operator.

The work connects finite differences, spectral gaps, and optimal transport.

Abstract

We consider probability measures supported on a finite discrete interval $[0,n]$. We introduce a new finitedifference operator $\nabla_n$, defined as a linear combination of left and right finite differences. We show that this operator $\nabla_n$ plays a key role in a new Poincar\'e (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.