It is known that if ( ) ∈ is a sequence of orthogonal polynomials in ([-1,1],()), then the roots are distributed according to an arcsine distribution (1 - ) for a wide variety of weights (). We connect this to a result of the Hilbert transform due to Tricomi: if ()(1 - ) ∈ (-1,1) and its Hilbert transform vanishes on (-1,1), then the function is a multiple of the arcsine distribution We also prove a localized Parseval-type identity that seems to be new: if ()(1- ) ∈ L(-1, 1) and has mean value 0 on (-1, 1), then .
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| http://dx.doi.org/10.1007/s00041-019-09678-w | DOI Listing |
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