Lower bounds for the Dyadic Hilbert transform

A REMARK ON THE ARCSINE DISTRIBUTION AND THE HILBERT TRANSFORM

It is known that if (p _n ) _n ∈_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)^1/4 ∈ 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 ) = c 1 - x 2 χ ( - 1 , 1 ) where c ∈ ℝ . We also prove a localized Parseval-type identity that seems to be new: if f(x)(1-x ²)^1/4 ∈ L²(-1, 1) and f ( x ) 1 - x 2 has mean value 0 on (-1, 1), then ∫ - 1 1 ( H f ) ( x ) 2 1 - x 2 d x = ∫ - 1 1 f ( x ) 2 1 - x 2 d x . .