Schur Parametrization of Positive Definite Block-Toeplitz Systems

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

doi:10.1137/0136004

Cited by 136 publications

(43 citation statements)

References 16 publications

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“…This proof makes essentially use of the Schur-Potapov algorithm for matrix-valued Schur functions which was done in [25] (see also [17,Section 3.8]) on the basis of the foregoing papers of Delsarte, Genin, and Kamp [8] and [9] on orthogonal matrix polynomials on T and related questions.…”

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confidence: 99%

The matricial Schur problem in both nondegenerate and degenerate cases

Fritzsche

Kirstein

Lasarow

2009

Mathematische Nachrichten

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Key wordsThe principal object of this paper is to present a new approach simultaneously to both nondegenerate and degenerate cases of the matricial Schur problem. This approach is based on an analysis of the central matrixvalued Schur functions which was started in [24]-[26] and then continued in [27]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well-known formula of D. Z. Arov and M. G. Kreȋn for that case (see [1]). Furthermore, we give some characterizations of the situation that the matricial Schur problem has a unique solution. The main goal of this paper is to present an approach to the matricial version of the classical Schur problem (see [32] and [33]) in both nondegenerate and degenerate cases. Roughly speaking, our strategy is due to an adaptation of the authors' former investigations to the general (i.e. possibly degenerate) matricial Carathéodory problem (see [29]). Our method is essentially based on the first and second authors former investigations [23]-[27] on the central matrix-valued Schur function associated with a finite Schur sequence of complex matrices. In particular, we will make frequently use of the matrix ball description of the elements of matricial Schur sequences. The main results of this paper (see Theorems 1.1, 3.2, 3.6, and 4.2) contain descriptions of the solution set of a matricial Schur problem in terms of a linear fractional transformation, the generating matrix-valued function of which is a matrix polynomial. The canonical blocks of this matrix polynomial will be constructed with the aid of those quadruple of matrix polynomials which were used in [27] to derive right and left quotient representations of central matrix-valued Schur functions (see Theorem 2.4).A different approach to the degenerate matricial Schur problem was used in [11]-[16] of V. K. Dubovoj. His method is based on a combination of V. P. Potapov's fundamental matrix inequality approach with the technique of subspaces of type K which was created by V. K. Dubovoj to handle the specific of the matricial Schur problem *

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confidence: 99%

The matricial Schur problem in both nondegenerate and degenerate cases

Fritzsche

Kirstein

Lasarow

2009

Mathematische Nachrichten

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“…Schur's classical algorithm delivers some coefficients (so-called Schur parameters) that turned out to be exactly the complex conjugates of the coefficients (so-called Szegő parameters) that appeared in the recurrence relations for the orthogonal polynomial as formulated by Szegő [24]. This classical result goes back to Geronimus (see [16] and for a matrix extension [11]). In view of the results presented below, we prove and apply a similar connection between an algorithm of Schur-Nevanlinna type and orthogonal rational functions on T (cf.…”

Section: Introductionmentioning

confidence: 91%

“…, λ m ∈ D which appear in the recurrence relations for the orthogonal rational functions (cf. [11] for the case of positive Hermitian block Toeplitz matrices). Since the matrix P is a crucial tool, we start in Section 2 with some basics on this generalized Schwarz-Pick matrix.…”

Section: Introductionmentioning

confidence: 99%

Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions

Bultheel¹,

Lasarow²

2008

Analysis

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Summary: An interpolation problem of Nevanlinna-Pick type for complex-valued Schur functions in the open unit disk is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points. Primarily, we study the case that there exist many Schur functions fulfilling the required conditions. For this situation, an application of the theory of orthogonal rational functions is used to characterize the set of all solutions of the problem in question. Moreover, we treat briefly the case of exactly one solution and present an explicit description of the unique solution in that case.

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“…The first one which was initiated by Djrbashian [9]- [12] and then later systematically worked out by Bultheel, González-Vera, Hendriksen, and Njåstad (see, e.g., [2]- [6]) is to study scalar orthogonal rational functions on the unit circle. The second generalization is to consider orthogonal matrix polynomials on the unit circle (see Delsarte/Genin/Kamp [7] and [8], Youla/Kazanjian [32], Fuhrmann [21], and [13,Section 3.6]). …”

Section: Introductionmentioning

confidence: 99%

Orthogonal rational matrix‐valued functions on the unit circle: Recurrence relations and a Favard‐type theorem

Fritzsche

Kirstein

Lasarow

2006

Mathematische Nachrichten

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Key wordsIn this paper, we combine both generalizations by studying orthogonal rational matrix-valued functions on the unit circle. In this way, we continue the line of investigations started in [18]-[20] where we realized first steps towards a Szegö theory for orthogonal rational matrix-valued functions on the unit circle.The main feature of our conception of this kind of Szegö theory is the distinguished role of the ChristoffelDarboux-formulas. A careful analysis of the scalar rational theory and the matrix polynomial case indicates that the Christoffel-Darboux formulas provide the key for extracting large amounts of essential information contained in orthonormal function systems. For this reason, we decided to place the Christoffel-Darboux formulas in the center of our conception. As a first cornerstone of our approach we introduced in [20, Section 6] the notions of left and right Christoffel-Darboux pairs of rational matrix-valued functions. The recursion formulas for left and right Christoffel-Darboux pairs which are obtained in Section 2 realize a crucial step in the way to our Szegö theory conception. Namely, these recursion formulas indicate that there is a one-to-one correspondence between Christoffel-Darboux pairs and sequences of matrices which are connected to particular signature matrices. In this way, we are led to study pairs of sequences of so-called j qq -recursively connected rational matrix-valued *

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Schur Parametrization of Positive Definite Block-Toeplitz Systems

Cited by 136 publications

References 16 publications

The matricial Schur problem in both nondegenerate and degenerate cases

The matricial Schur problem in both nondegenerate and degenerate cases

Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions

Orthogonal rational matrix‐valued functions on the unit circle: Recurrence relations and a Favard‐type theorem

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