Fine structure of the zeros of orthogonal polynomials, I. A tale of two pictures

Barry Simon

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: $\alpha_n =Cb^n + O((b\Delta)^n)$. 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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Key words

OPUC, clock behavior, Poisson zeros, orthogonal polynomials

AMS subject classifications

42C05, 30C15, 60G55