LAUSR

It is known that if $$(p_n)_{n \in \mathbb {N}}$$(pn)n∈N is a sequence of orthogonal polynomials in $$L^2([-1,1], w(x)dx)$$L2([-1,1],w(x)dx), then the roots are distributed according to an arcsine distribution $$\pi ^{-1} (1-x^2)^{-1}dx$$π-1(1-x2)-1dx for a wide variety of weights w(x). We connect this to a result of the Hilbert transform due to Tricomi: if $$f(x)(1-x^2)^{1/4} \in L^2(-1,1)$$f(x)(1-x2)1/4∈L2(-1,1) and its Hilbert transform Hf vanishes on $$(-1,1)$$(-1,1), then the function f is a multiple of the arcsine distribution $$\begin{aligned} f(x) = \frac{c}{\sqrt{1-x^2}}\chi _{(-1,1)} \qquad \text{ where }~c~\in \mathbb {R}. \end{aligned}$$f(x)=c1-x2χ(-1,1)wherec∈R.We also prove a localized Parseval-type identity that seems to be new: if $$f(x)(1-x^2)^{1/4} \in L^2(-1,1)$$f(x)(1-x2)1/4∈L2(-1,1) and $$f(x) \sqrt{1-x^2}$$f(x)1-x2 has mean value 0 on $$(-1,1)$$(-1,1), then $$\begin{aligned} \int _{-1}^{1}{ (Hf)(x)^2 \sqrt{1-x^2} dx} = \int _{-1}^{1}{ f(x)^2 \sqrt{1-x^2} dx}. \end{aligned}$$∫-11(Hf)(x)21-x2dx=∫-11f(x)21-x2dx.