A Remark on the Arcsine Distribution and the Hilbert Transform

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 $f(x)(1-x^2{)}^{1\mathrm{/}4}\in L^2(-1,1)$ and its Hilbert transform $Hf$ vanishes on $(-1,1)$, then the function $f$ is a multiple of the arcsine distribution $$f(x)=\frac{c}{\sqrt{1-x^2}}{\chi }_{(-1,1)}wherec\in \mathbb{R}\mathrm{.}$$ We also prove a localized Parseval-type identity that seems to be new: if $f(x)(1-x^2{)}^{1\mathrm{/}4}\in L^2(-1,1)$ and $f(x) \sqrt{1-x^2}$hasmeanvalue0on$(-1,1)$,then$$ \int_{-1}^{1}{ (Hf)(x)^2 \sqrt{1-x^2} dx} = \int_{-1}^{1}{ f(x)^2 \sqrt{1-x^2} dx}.$$