Orthogonal polynomials on the unit circle: New results

By Barry Simon. 2004

460 citations recorded by the source corpus.

We announce numerous new results in the theory of orthogonal polynomials on the unit circle, most of which involve the connection between a measure on the unit circle in the complex plane and the coefficients in the recursion relations for the polynomials known as Verblunsky coefficients. Included are several applications of the recently discovered matrix realization of Cantero, Moral, and Velazquez. In analogy with the spectral theory of Jacobi matrices, several classes of exotic Verblunsky coefficients are studied. A version of Rahkmanov's theorem is proven with a single gap with eigenvalues allowed in the gap. Analogs of Borg's theorem and the Birman-Schwinger principle are found.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/20331f11b318a163bf4b2b21b68f5bd98eb046a2

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In Applied Mathematics

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Citations in the corpus

A New Approach to Universality Limits Involving Orthogonal Polynomials (2007, methodology): “Throughout we use w = dµ dx to denote the absolutely continuous part of µ.”
Recent Progress on Truncated Toeplitz Operators (2011, background): “ptotic analysis of Toeplitz operators on H2 via large truncated Toeplitz matrices is the focus of [19]. The role played by Toeplitz determinants in the study of orthogonal polynomials is discussed in [93,94] and its relationship to random matrix theory is examined in [15,69]. Finally, we should also”
Christoffel functions on curves and domains (2009, background): “Their asymptotics on the real line and on the unit circle have been well understood ([34], [17], [18], [36], [6], [29], [32] [31], [16], [4]), but nothing was known about their asymptotic behavior on general curves (for a subarc of the circle and some special measures see [6]).”
Two extensions of Lubinsky’s universality theorem (2008, background): “This paper primarily discusses orthogonal polynomials on the real line (OPRL) [39, 10, 28].”
EQUILIBRIUM MEASURES AND CAPACITIES IN SPECTRAL THEORY (2007, background): “anμ(R) and if supp(dμ) ⊂ ∂D, the unit circle, we use Φn, φn and note (see [108, 44, 98, 99]) there are Verblunsky coefficients {αn}n=0 ∈ D∞, so Φn+1(z) = zΦn − ᾱnΦn(z) (1.”