Fine Structure of the Zeros of Orthogonal Polynomials, I. A Tale of Two Pictures

2004/11/17 by Barry Simon, Simon, Barry · 1 citation
Mathematics · #42C05 #FOS: Mathematics #Primary 4C05 #Secondary 47B36 #Spectral Theory (math.SP) #math.SP #msc:42C05 #msc:47B36 #msc:4C05

paper · pdf · doi:10.48550/arxiv.math/0411391

Keywords: orthogonal polynomials, Jacobi matrices, CMV matrices

arxiv created 2004/11/17 · arxiv updated 2009/12/01

Abstract

Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large n. Motivated by two plots, we look at the finer structure for the case of random Verblunsky coefficients and for what we call the BLS condition: α_n = Cbⁿ + O((bΔ)ⁿ). In the former case, we describe results of Stoiciu. In the latter case, we prove asymptotically equal spacing for the bulk of zeros.

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• Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients

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