On the “Favard theorem” and its extensions

Marcellán, Francisco; Álvarez-Nodarse, R.

doi:10.1016/s0377-0427(00)00497-0

Cited by 46 publications

(39 citation statements)

References 18 publications

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“…In agreement with classical literature, we refer to this result as Favard's theorem (see, among others, [Chihara 1978;Erdélyi et al 1991;Favard 1935;Marcellán and Álvarez-Nodarse 2001;Szegő 1975]) or the spectral theorem [Ismail 2005], even though this result is previously contained in the works of Stieltjes [1895;1894] and Stone [1932]. In the positive definite case, an analog of Favard's theorem for OPUC based on the construction of a sequence of absolutely continuous measures whose limit is the spectral measure is presented in [Erdélyi et al 1991].…”

Section: Recurrence Relation and Spectral Theoremsupporting

confidence: 86%

“…In this paper the authors follow a method used previously by Delsarte, Genin and Kamp [Delsarte et al 1978] who consider the matrix-valued case. Some extensions to the quasidefinite case have been analyzed in [Marcellán and Álvarez-Nodarse 2001]. Our proof follows a standard scheme (constructive approach) which goes back at last to [Chihara 1978]; see also [Marcellán and Álvarez-Nodarse 2001].…”

Section: Recurrence Relation and Spectral Theoremmentioning

confidence: 99%

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On a spectral theorem in paraorthogonality theory

2016

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Motivated by the works of Genin (1988, 1991), who studied paraorthogonal polynomials associated with positive definite Hermitian linear functionals and their corresponding recurrence relations, we provide paraorthogonality theory, in the context of quasidefinite Hermitian linear functionals, with a recurrence relation and the analogous result to the classical Favard's theorem or spectral theorem. As an application of our results, we prove that for any two monic polynomials whose zeros are simple and strictly interlacing on the unit circle, with the possible exception of one of them which could be common, there exists a sequence of paraorthogonal polynomials such that these polynomials belong to it. Furthermore, an application to the computation of Szegő quadrature formulas is also discussed.

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Section: Recurrence Relation and Spectral Theoremsupporting

confidence: 86%

Section: Recurrence Relation and Spectral Theoremmentioning

confidence: 99%

On a spectral theorem in paraorthogonality theory

2016

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“…Theorem 4 (Favard [16]): A monic polynomial sequence {Pn(x)} is an orthogonal polynomial sequence if it satisfies a three-term recurrence relation…”

Section: Appendixmentioning

confidence: 99%

Stabilizing Quantum States by Constructive Design of Open Quantum Dynamics

Ticozzi

Schirmer

Wang

2010

IEEE Trans. Automat. Contr.

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Abstract-Based on recent work on the asymptotic behavior of controlled quantum Markovian dynamics, we show that any generic quantum state can be stabilized by devising constructively a simple Lindblad-GKS generator that can achieve global asymptotic stability at the desired state. The applicability of such result is demonstrated by designing a direct feedback strategy that achieves global stabilization of a qubit state encoded in a noise-protected subspace.

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“…There exists a Hermitian functional (matrix valued) on M[x] with respect to which the polynomials are orthogonal, in the spirit of Chihara's extension of Favard's theorem to complex values, see [5], also [17]:…”

Section: Commutative Families Of Polynomialsmentioning

confidence: 99%

Matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas

Costin

2009

Journal of Approximation Theory

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The properties of matrix-valued polynomials generated by the scalar-type Rodrigues' formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the commutative case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators.A new, direct proof is given to the conjecture of Durán and Grünbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type.Commutative classes of orthogonal polynomials (corresponding to weights that are self-adjoint but not positive semidefinite) are found, which satisfy all the properties usually associated to orthogonal polynomials, and are not of scalar type.

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On the “Favard theorem” and its extensions

Abstract: In this paper we present a survey on the "Favard theorem" and its extensions.

Cited by 46 publications

References 18 publications

On a spectral theorem in paraorthogonality theory

On a spectral theorem in paraorthogonality theory

Stabilizing Quantum States by Constructive Design of Open Quantum Dynamics

Matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas

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