Orthogonal polynomial matrices on the unit circle

Delsarte, P.; Genin, Y.; Kamp, Y.

doi:10.1109/tcs.1978.1084452

Cited by 199 publications

(136 citation statements)

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“…A Hermitian integrable q×q matrix function W on T is called a weight q×q matrix if W ≥ 0 and det W(e it ) ̸ = 0 holds almost everywhere [5]. Let us suppose that W is a weight q × q matrix on T. The space L 2 (W) is defined by…”

Section: πmentioning

confidence: 99%

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On a generalization of the maximum entropy Theorem of Burg

Marcano

Infante

Sánchez

2017

RMTA

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In this article we introduce some matrix manipulations that allow us to obtain a version of the original Christoffel-Darboux formula, which is of interest in many applications of linear algebra. Using these developments matrix and Jensen's inequality, we obtain the main result of this proposal, which is the generalization of the maximum entropy theorem of Burg for multivariate processes.Keywords: multivariate processes; maximum entropy theorem; Christoffel-Darboux formula. ResumenEn este artículo se introducen algunas manipulaciones matriciales que nos permiten obtener una versión de la fórmula original de ChristoffelDarboux, la cual resulta de interés en muchas aplicaciones del álgebra lineal. Usando estos desarrollos matriciales y la desigualdad de Jensen, se obtiene el resultado principal de esta propuesta, que consiste en la generalización del teorema de entropía máxima de Burg para procesos multivariados.Palabras clave: procesos multivariados; teorema de entropía máxima; fórmula de Christoffel-Darboux.Mathematics Subject Classification: 94A17, 46E22.

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Section: πmentioning

confidence: 99%

“…In the third section we obtain a matrix version of the Christoffel-Darboux formula (cf. [5]). In the last section, we state the main result of the paper.…”

Section: Introductionmentioning

confidence: 99%

On a generalization of the maximum entropy Theorem of Burg

Marcano

Infante

Sánchez

2017

RMTA

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“…is considered, see also [10], [45]. The result (19) constitutes a generalization towards block-polynomials of a data-dependent discrete inner product for vector-polynomials, see also [7, Sect.…”

Section: Selection Of a Data-dependent Polynomial Basismentioning

confidence: 99%

Bi-Orthonormal Polynomial Basis Function Framework With Applications in System Identification

Herpen

Bosgra

Oomen

2016

IEEE Trans. Automat. Contr.

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Abstract-Numerical aspects are of central importance in identification and control. Many computations in these fields involve approximations using polynomial or rational functions that are obtained using orthogonal or oblique projections. The aim of this paper is to develop a new and general theoretical framework to solve a large class of relevant problems. The proposed method is built on the introduction of bi-orthonormal polynomials with respect to a data-dependent bi-linear form. This bi-linear form generalises the conventional inner product and allows for asymmetric and indefinite problems. The proposed approach is shown to lead to optimal numerical conditioning (κ = 1) in a recent frequency-domain instrumental variable system identification algorithm. In comparison, it is shown that these recent algorithms exhibit extremely poor numerical properties when solved using traditional approaches. I. INTRODUCTIONApplications of system identification and control often involve numerical computations. The accuracy of these computations determines the quality of the resulting model or controller. This has led to considerable research to develop numerically reliable algorithms, see, e.g., [38] and [8], [4] for a general overview. Many computations involve weighted least-squares type problems for systems that are parametrized in terms of (vector) polynomials or rational forms, where orthogonal or oblique projections play a central role.In the field of system identification, weighted nonlinear least-squares criteria are particularly common, see [29]. Established identification algorithms include [26], the SK-iteration [34], and the Gauss-Newton iteration [1], which (iteratively) compute the least-squares solution to a linear systems of equations, for polynomial models or rational parametrizations. Although conceptually straightforward, the associated numerical conditioning is often extremely poor, as is evidenced by the developments in [1 , a fundamentally different solution strategy is pursued. The strategy is based upon the construction of orthogonal polynomials with respect to a data-dependent inner product, which directly provides the solution to the approximation problem in terms of polynomials. Essentially, this yields optimal conditioning of the associated linear system of equations, i.e., κ = 1.Besides developments in view of reliable algorithms that involve least-squares type solutions, recently, more general

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“…The proof of (17) is essentially the same as in §2. It is based on the RieszHerglotz representation (7) for nondegenerate matrix-valued Carathéodory functions and makes use of a suitable matrix extension of the Schwarz inequality [2]. Details will not be given.…”

Section: Generalizationmentioning

confidence: 99%