A Remark on the Arcsine Distribution and the Hilbert Transform

TL;DR

This paper explores the relationship between orthogonal polynomial roots, the arcsine distribution, and the Hilbert transform, establishing new identities and characterizations of functions related to these concepts.

Contribution

It connects the distribution of polynomial roots to the Hilbert transform and introduces a new localized Parseval-type identity involving these functions.

Findings

Roots of orthogonal polynomials follow the arcsine distribution.

Functions with vanishing Hilbert transform are multiples of the arcsine distribution.

A new localized Parseval identity for functions related to the arcsine distribution.

Abstract

It is known that if $(p_n)_{n \in \mathbb{N}}$ is a sequence of orthogonal polynomials in $L^2([-1,1], w(x)dx)$, then the roots are distributed according to an arcsine distribution $\pi^{-1} (1-x^2)^{-1}dx$ for a wide variety of weights $w(x)$. We connect this to a result of the Hilbert transform due to Tricomi: if $f(x)(1-x^2)^{1/4} \in L^2(-1,1)$ and its Hilbert transform $Hf$ vanishes on $(-1,1)$, then the function $f$ is a multiple of the arcsine distribution $$ f(x) = \frac{c}{\sqrt{1-x^2}}\chi_{(-1,1)} \qquad \mbox{where}~c~\in \mathbb{R}.$$ We also prove a localized Parseval-type identity that seems to be new: if $f(x)(1-x^2)^{1/4} \in L^2(-1,1)$ and $f(x) \sqrt{1-x^2}$ has mean value 0 on $(-1,1)$, then $$ \int_{-1}^{1}{ (Hf)(x)^2 \sqrt{1-x^2} dx} = \int_{-1}^{1}{ f(x)^2 \sqrt{1-x^2} dx}.$$