Second degree functionals on the unit circle

Cachafeiro, A.; Pérez, Castillo

doi:10.1080/10652460410001673004

Cited by 4 publications

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References 3 publications

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“…If B ≡ 0 and C, D specific polynomials, we obtain the semi-classical class on the unit circle (see [2,6,11]). Moreover, the Laguerre-Hahn class on the unit circle includes the class of second degree functionals on 2 BRANQUINHO AND REBOCHO the unit circle (see [4]) and includes, as we will see on section 3, the linearfractional transformations of Carathéodory functions which are Laguerre-Hahn.…”

Section: Introductionmentioning

confidence: 99%

Distributional equation for Laguerre–Hahn functionals on the unit circle

Branquinho

Rebocho

2009

Journal of Computational and Applied Mathematics

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Let u be a hermitian linear functional defined in the linear space of Laurent polynomials and F its corresponding Carathéodory function. We establish the equivalence between a Riccati differential equation with polynomial coefficients for F , zAF ′ = BF 2 + CF + D and a distributional equation for u, D(Au) = B 1 u 2 + C 1 u + H 1 L, where L is the Lebesgue functional, and the polynomials B 1 , C 1 , D 1 are defined in terms of the polynomials A, B, C, D .

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Section: Introductionmentioning

confidence: 99%

Distributional equation for Laguerre–Hahn functionals on the unit circle

Branquinho

Rebocho

2009

Journal of Computational and Applied Mathematics

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The Euler–Lagrange theory for Schur's Algorithm: Algebraic exposed points

Khrushchev

2006

Journal of Approximation Theory

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In this paper the ideas of Algebraic Number Theory are applied to the Theory of Orthogonal polynomials for algebraic measures. The transferring tool are Wall continued fractions. It is shown that any set of closed arcs on the circle supports a quadratic measure and that any algebraic measure is either a Szegö measure or a measure supported by a proper subset of the unit circle consisting of a finite number of closed arcs. Singular parts of algebraic measures are finite sums of point masses.

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Characterizing orthogonality measures on the bounded interval¹

Cachafeiro

Pérez

2008

Integral Transforms and Special Functions

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The aim of this paper is to characterize a new class of measures on the bounded interval, which contains the semiclassical class. The characterizations are given in terms of the differential properties of the Stieltjes function and also in terms of the differential properties of the formal series of moments associated with the Chebyshev basis of first kind. We also present several interesting examples and we obtain the differential equations satisfied by the family of orthogonal polynomials when the measure belongs to the Szegö class.

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Second degree functionals on the unit circle

Cited by 4 publications

References 3 publications

Distributional equation for Laguerre–Hahn functionals on the unit circle

Distributional equation for Laguerre–Hahn functionals on the unit circle

The Euler–Lagrange theory for Schur's Algorithm: Algebraic exposed points

Characterizing orthogonality measures on the bounded interval¹

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Second degree functionals on the unit circle

Cited by 4 publications

References 3 publications

Distributional equation for Laguerre–Hahn functionals on the unit circle

Distributional equation for Laguerre–Hahn functionals on the unit circle

The Euler–Lagrange theory for Schur's Algorithm: Algebraic exposed points

Characterizing orthogonality measures on the bounded interval1

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Characterizing orthogonality measures on the bounded interval¹