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

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: ${\alpha }_n=C{b}^n+O((b\Delta {)}^n)$. In the former case, we describe results of Stoiciu. In the latter case, we prove asymptotically equal spacing for the bulk of zeros.

Keywords: orthogonal polynomials, Jacobi matrices, CMV matrices

Cited by in corpus (5)

• Fine Structure of the Zeros of Orthogonal Polynomials, II. OPUC With Competing Exponential Decay
• Szego polynomials: a view from the Riemann-Hilbert window
• Meromorphic Szego functions and asymptotic series for Verblunsky coefficients
• Fine structure of the zeros of orthogonal polynomials, III. Periodic recursion coefficients
• The Statistical Distribution of the Zeros of Random Paraorthogonal Polynomials on the Unit Circle