Connection between orthogonal polynomials on the unit circle and bounded interval

Berriochoa, E.; Cachafeiro, A.; García-Amor, J.

doi:10.1016/j.cam.2004.09.018

Cited by 15 publications

(18 citation statements)

References 3 publications

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“…Proof of Proposition 3 Expressions (16), (17), (19) and (20) are obtained expressing the factorization problem as a system of equations. From (17) we deduce (S 1 ) lk = (S 2 ) ll ((g [l+1] ) −1 ) l,k = (S 2 ) ll det g [l+1] (−1) l+k M (l+1) k,l , so that ϕ (l)…”

Section: Olpuc Fulfillmentioning

confidence: 99%

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

Álvarez-Fernández¹,

Mañas²

2013

Advances in Mathematics

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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-Velázquez 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-Velázquez 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 Szegő 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 τ -functions is presented and bilinear equations are derived.

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Section: Olpuc Fulfillmentioning

confidence: 99%

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

Álvarez-Fernández¹,

Mañas²

2013

Advances in Mathematics

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“…In order to obtain these results first we recall the connection between measures supported on [−π , π] and [−1, 1] respectively through the following four transformations (see [16,11,15]). …”

Section: Quadrature Rules For Polynomial Modifications Of the Bernstementioning

confidence: 99%

“…The results are obtained using the so-called Szegő transformation and its generalizations, that is, the relations between measures supported on the interval [−1, 1] and symmetric measures supported the unit circle T = {z : |z| = 1} (see [15,16]), and applying the quadrature formula for the polynomial modification of the Bernstein-Szegő measures that we have proved in Section 2.…”

Section: Quadrature Rules For Polynomial Modifications Of the Bernstementioning

confidence: 99%

Characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials

Berriochoa

Cachafeiro

Amor

2009

Computers & Mathematics with Applications

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a b s t r a c tIn the present paper we characterize the measures on the unit circle for which there exists a quadrature formula with a fixed number of nodes and weights and such that it exactly integrates all the polynomials with complex coefficients. As an application we obtain quadrature rules for polynomial modifications of the Bernstein measures on [−1, 1], having a fixed number of nodes and quadrature coefficients and such that they exactly integrate all the polynomials with real coefficients.

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“…, n − 1 ( [5] and also [14]). Orthogonal polynomials on the real line with support on [−1, 1] are connected with the Szegő polynomials [15,16]. The extension to the unit circle context of the three-term relations and tridiagonal Jacobi matrices of the real scenario, require of Hessenberg matrices and give the Szegő recursion relation, which is expressed in terms of the reciprocal, or reverse, Szegő polynomials P * l (z) := z l P l (z −1 ) and reflection or Verblunsky coefficients α l := P l (0), ( P l P * l ) = ( z α l zᾱ l 1 )( P l−1 P * l−1 ).…”

Section: Introductionmentioning

confidence: 99%

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

Ariznabarreta¹,

Mañas²,

Toledano³

2018

Stud Appl Math

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Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities such as Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied by a Laurent polynomial, or are multiplied by the complex conjugate of a Laurent polynomial. For the Geronimus transformation, the linear functional is perturbed in two possible manners as well, by a division by a Laurent polynomial or by a complex conjugate of a Laurent polynomial, in both cases the addition of appropriate masses (linear functionals supported on the zeros of the perturbing Laurent polynomial) is considered. The connection formulas for the CMV biorthogonal Laurent polynomials, its norms, and Christoffel-Darboux kernels, in all the four cases, are given. For the Geronimus transformation, the connection formulas for the second kind functions and mixed Christoffel-Darboux kernels are also given in the two possible cases. For prepared Laurent polynomials, i.e., of the form L(z) = L n z n + · · · + L −n z −n , L n L −n = 0, these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas for all the four transformations, expressing an arbitrary degree perturbed biorthogonal Laurent polynomial in terms of 2n unperturbed biorthogonal Laurent polynomials, their second kind functions or Christoffel-Darboux kernels and its mixed versions. Different curves are presented as examples, such as the real line, the circle, the Cassini oval,

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Connection between orthogonal polynomials on the unit circle and bounded interval

Cited by 15 publications

References 3 publications

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

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

Characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

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