Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle

By Barry Simon. 2006

50 citations recorded by the source corpus.

We prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators. In particular, we obtain new details on the interlacing of zeros for successive POPUC.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/29c297c748d477d7fd970ef93813d163d51ed60f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Yoram Last
2. Barry Simon
3. David Damanik
4. Doron S. Lubinsky
5. N. H. Bingham
6. Alexander Pushnitski
7. David Damanik
8. Vilmos Totik

In Mathematical Physics

1. Yoram Last
2. Barry Simon
3. Doron S. Lubinsky
4. David Damanik
5. David Damanik
6. Alexander Pushnitski
7. Vilmos Totik

In Statistics and Probability

1. Yoram Last
2. Barry Simon

Citations in the corpus

• Fine Structure of the Zeros of Orthogonal Polynomials, IV. A Priori Bounds and Clock Behavior (2006, background): “s of ˜ϕn lie on ∂D; indeed, they are the zeros of a finite unitary matrix (see Theorem 8.2.7 of [39]). Zeros of POPUC are discussed extensively in Golinskii [14], Cantero-MoralVel´azquez [3, 4], Simon [46], and Wong [55]. As explained in Section 10.8 of [40], there is an OPUC analog of (2.7)–(2.8) (n”
• The Christoffel-Darboux Kernel (2008, methodology): “We were motivated to look for this by a paper of Wong [105], who derived the basics for zeros of POPUC (paraorthogonal polynomials on the unit circle) using the CD formula (for other approaches to zeros of POPUCs, see [19, 86]).”
• The Analytic Theory of Matrix Orthogonal Polynomials (2007, unknown): “8 Matrix POPUC Analogously to the scalar case (see [15, 100, 119, 172, 197]), given any unitary β in Ml, we define φn (z;β) = zφ R n−1(z) − φ n−1(z)β.”