A Remark on the Arcsine Distribution and the Hilbert Transform

Abstract

We prove that 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 \[ f(x) = \frac{c}{\sqrt{1-x^2}}\chi_{(-1,1)} \qquad \mbox{where}~c~\in \mathbb{R}.\] This characterization of the arcsine distribution explains why roots of orthogonal polynomials tend to follow an arcsine distribution at a great level of generality. Conversely, 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 \[ \int_{-1}^{1}{ (Hf)(x)^2 \sqrt{1-x^2} dx} = \int_{-1}^{1}{ f(x)^2 \sqrt{1-x^2} dx}.\]