A Remark on the Arcsine Distribution and the Hilbert Transform

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 $$ f(x) = \frac{c}{\sqrt{1-x^2}}χ_{(-1,1)} \qquad \mbox{where}~c~\in \mathbb{R}.$$ 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 $${\int }_{-1}^1(Hf)(x{)}^2\sqrt{1-x^2}dx={\int }_{-1}^{1}f(x{)}^2\sqrt{1-x^2}dx.$$

The Isometry property was derived previously by Ledoux & Popescu (The One Dimensional Free Poincare Inequality)