A REMARK ON THE ARCSINE DISTRIBUTION AND THE HILBERT TRANSFORM

It is known that if (p ) ∈_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 π ^-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 ∈ L ²(-1,1) and its Hilbert transform Hf vanishes on (-1,1), then the function f is a multiple of the arcsine distribution We also prove a localized Parseval-type identity that seems to be new: if f(x)(1-x ²)¹ ∈ L²(-1, 1) and $$ has mean value 0 on (-1, 1), then .