Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

By María José Cantero, Leandro Moral, Luis Velázquez. 2002

303 citations recorded by the source corpus.

It is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five-diagonal matrices depending on the Schur parameters. This result is achieved through the study of orthogonal Laurent polynomials on the unit circle. More precisely, it is a consequence of the five term recurrence relation obtained for these orthogonal Laurent polynomials, and the one to one correspondence established between them and the orthogonal polynomials on the unit circle. As an application, some results relating the behaviour of the zeros of orthogonal polynomials and the location of Schur parameters are obtained.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f0e93a22bb7e4e0d53d30236c0a2fbb7449b921e

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Citations in the corpus

Matrix models for circular ensembles (2004, background): “(5.12) We do not have a direct derivation of this fact.”
Orthogonal polynomials on the unit circle: New results (2004, unknown): “Then Φ n has at most one zero in {z | |z − z 0 | < δ/3}.”
Szego Orthogonal Polynomials with Respect to an Analytic Weight: Canonical Representation and Strong Asymptotics (2005, methodology): “The m ain technique isthe steepestdescentanalysisofDeiftand Zhou,based on them atrix Riem ann-Hilbert characterization proposed by Fokas,Itsand K itaev.”
CMV matrices: Five years after (2006, background): “18) As realized in [12], the { n}∞n=0 and {xn}∞n=0 are always a basis of L2( D, d ) since the Laurent polynomials are dense on C( D), while { n}∞n=0 may or may not be a basis (it is known that this is a basis if and only if ∑ n | n|2 =∞; see [69, Theorem 1.”
OPUC on one foot (2005, background): “The right basis, as discovered by Cantero, Moral, and Velázquez [14] is the one, χ0, χ1, χ2, . . . , obtained by orthonormalizing 1, z, z −1, z2, z−2, . . . .”
FINE STRUCTURE OF THE ZEROS OF ORTHOGONAL POLYNOMIALS, I. A TALE OF TWO PICTURES (2004, methodology): “Mhaskar-Saff complete the proof by using potential theory to analyze the limit points of the . Instead, I will sketch a different idea from [36, Section 8.2] that exploits the CMV matrix (see [ 5 ] and [36, Sections 4.2n4.5]).”
Minimal representations of unitary operators and orthogonal polynomials on the unit circle (2004, background): “Hence, the referred relation between eigenvectors is just a consequence of the relation (4) between OP and OLP.”
Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective (2009, background): “) (the so-called bipartite Coxeter element) gives rise to a special kind of pentadiagonal matrices X (called CMV matrices, after Cantero, Moral and Velázquez), which serve as an alternative version of recursion operators for orthogonal polynomials on the unit circle, see [6] and [32].”
Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles (2009, background): “The temperature β−1 appears as the square of the coupling constant.”