Inequalities for Jacobi polynomials

TL;DR

This paper establishes a uniform Bernstein type inequality for Jacobi polynomials applicable across all degrees and parameters, providing bounds on matrix coefficients for SU(2) representations and extending previous mathematical results.

Contribution

It introduces a new uniform inequality for Jacobi polynomials that applies broadly and enhances understanding of matrix coefficients in representation theory.

Findings

Derived a Bernstein type inequality valid for all degrees and parameters.

Provided bounds with decay rate $d^{-1/4}$ for matrix coefficients of SU(2).

Extended previous results related to a conjecture by Erdélyi, Magnus, and Nevai.

Abstract

A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of matrix coefficients for the irreducible representations of $\mathrm{SU}(2)$ with a decay of $d^{-1/4}$ in the dimension $d$ of the representation. Moreover it complements previous results of Krasikov on a conjecture of Erd\'elyi, Magnus and Nevai.