Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

Let there be given a probability measure µ on the unit circle T of the complex plane and consider the inner product induced by µ. In this paper we consider the problem of orthogonalizing a sequence of monomials {zrk}k, for a certain order of the rk ∈ Z, by means of the Gram-Schmidt orthogonalization process. This leads to a basis of orthonormal Laurent polynomials {ψk}k. We show that the matrix representation with respect to the basis {ψk}k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped ’ matrix factorization. Here the ‘snake shape ’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {zrk}k are orthogonalized, while the ‘segments ’ of the snake are canonically determined in terms of the Schur parameters for µ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism

Orthogonal Laurent polynomials in unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

Orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss--Borel factorization of a Cantero-Moral-Velazquez moment matrix, which is constructed in terms of a complex quasi-definite measure supported in the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials in the unit circle and the corresponding second kind functions. Jacobi operators, 5-term recursion relations and Christoffel-Darboux kernels, projecting to particular spaces of truncated Laurent polynomials, and corresponding Christoffel-Darboux formulae are obtained within this point of view in a completely algebraic way. Cantero-Moral-Velazquez sequence of Laurent monomials is generalized and recursion relations, Christoffel-Darboux kernels, projecting to general spaces of truncated Laurent polynomials and corresponding Christoffel-Darboux formulae are found in this extended context. Continuous deformations of the moment matrix are introduced and is shown how they induce a time dependant orthogonality problem related to a Toda-type integrable system, which is connected with the well known Toeplitz lattice. Using the classical integrability theory tools the Lax and Zakharov-Shabat equations are obtained. The dynamical system associated with the coefficients of the orthogonal Laurent polynomials is explicitly derived and compared with the classical Toeplitz lattice dynamical system for the Verblunsky coefficients of Szeg\H{o} polynomials for a positive measure. Discrete flows are introduced and related to Darboux transformations. Finally, the representation of the orthogonal Laurent polynomials (and its second kind functions), using the formalism of Miwa shifts, in terms of $\tau$-functions is presented and bilinear equations are derived

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

of Toda-like integrable systems are connected us-ing the Gauss-Borel factorization of two, left and a right, Cantero-Morales-Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. Ablock Gauss-Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szegő polynomials, which can be expressed in terms of Schur complements of bordered trun-cations of the block moment matrix. The corresponding block extension of the Christoffel-Darboux theory is derived. De-formations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov-Shabat equations, bilinear equa-tions and discrete flows-connected with Darboux transformations. We generalize the integrable flows of the Cafasso’s matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szegő polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel-Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomialsM.M. thanks economical support from the Spanish “Ministerio de Economía y Competitividad” research project MTM2012-36732-C03-01, Ortogonalidad y aproximación; teoría y aplicaciones

Polinomios biortogonales y sus generalizaciones: una perspectiva desde los sistemas integrables

La conexión existente entre los polinomios ortogonales y otras ramas de la matemática, la física o la ingeniería es verdaderamente asombrosa. Además, no hay mejor prueba de la utilidad de estos que el propio crecimiento, avance perpetuo y generalización en diversas direcciones de lo que se entendía por polinomio ortogonal en los albores de la teoría. Conforme el concepto se fue generalizando, también fueron evolucionando las técnicas para su estudio, algunas de estas claramente influenciadas por aquellas disciplinas matemáticas con las que iban surgiendo conexiones. La perspectiva que esta tesis adopta frente a los polinomios ortogonales es un ejemplo de este tipo de influencias, compartiendo herramientas y entrelazandose con la teoría de los sistemas integrables. Una posición privilegiada en esta tesis la ocuparían las matrices de Gram semi in nitas; cada cual asociada a una forma sesquilineal adaptada al tipo de biortogonalidad en cuestión. A estas matrices se les impondrán una serie de condiciones cuyo objeto sería el de garantizar la existencia y unicidad de las secuencias biortogonales asociadas a las mismas. El siguiente paso consistiría en buscar simetrías de estas matrices de Gram. Existen dos razones por las que este esfuerzo resulta ventajoso. En primer lugar, cada simetría encontrada podría traducirse en propiedades de las secuencias biortogonales, por ejemplo: una estructura Hankel de la matriz es equivalente a gozar de la recurrencia a tres términos de los polinomios ortogonales; la simetría propia de las matrices asociadas a pesos clásicos (Hermite, Laguerre, Jacobi) implica la existencia del operador diferencial lineal de segundo orden de que los polinomios clásicos son solución; etc..