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

$$\begin{aligned} f(x) = \frac{c}{\sqrt{1-x^2}}\chi _{(-1,1)} \qquad \text{ where }~c~\in \mathbb {R}. \end{aligned}$$

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

$$\begin{aligned} \int _{-1}^{1}{ (Hf)(x)^2 \sqrt{1-x^2} dx} = \int _{-1}^{1}{ f(x)^2 \sqrt{1-x^2} dx}. \end{aligned}$$