Jacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator

TL;DR

This paper establishes a novel connection between Bernstein-type inequalities for Jacobi polynomials and dispersive estimates for Schrödinger equations with the generalized Laguerre operator, leading to new mathematical bounds.

Contribution

It introduces a new link between polynomial inequalities and dispersive PDE estimates, utilizing known bounds to derive novel results for Schrödinger equations.

Findings

Dispersive estimates for Schrödinger equations with Laguerre operators are connected to Bernstein inequalities for Jacobi polynomials.

New dispersive decay estimates are derived from known polynomial bounds.

These results lead to the establishment of new Bernstein-type inequalities.

Abstract

The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schr\"odinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schr\"odinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.