The Schur Algorithm for Generalized Schur Functions IV: Unitary Realizations

Alpay, Daniel; Azizov, T. Ya.; Dijksma, Aad; Langer, H.; Wanjala, Gerald

doi:10.1007/978-3-0348-7881-4_2

Cited by 11 publications

(9 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The essential tool was the theory of reproducing kernel Pontryagin spaces and the Schur algorithm for generalized Schur functions as developed in [2][3][4][5][6][7]12,14,17,19].…”

Section: Introductionmentioning

confidence: 99%

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

Alpay

Dijksma

Langer

2004

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…The essential tool was the theory of reproducing kernel Pontryagin spaces and the Schur algorithm for generalized Schur functions as developed in [2][3][4][5][6][7]12,14,17,19].…”

Section: Introductionmentioning

confidence: 99%

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

Alpay

Dijksma

Langer

2004

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…Azizov, A. Dijksma, H. Langer and G. Wanjala (see [30][31][32][33][34]). Now we will sketch some recent developments on multivariable analogues of the Schur class in the unit disk.…”

Section: On Some Generalizations Of Schur Functions and The Classicalmentioning

confidence: 97%

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

Bogner

Fritzsche

Kirstein

2006

Complex anal.oper.theory

View full text Add to dashboard Cite

This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first to develop a number of applications. A central theme of this paper is to study the interplay between functions from Sp×q,∞(D) and their sequence of SP-parameters. In particular, we describe how certain summability properties of the SP-parameters are expressed in terms of the associated functions from Sp×q,∞(D). As a byproduct of our investigations on the interplay between the SP-algorithm for p × q Schur functions and the SP-algorithm for sequences of complex p×q matrices we present a new approach to the nondegenerate matricial Schur problem. Our method complies with the basic strategy in Schur's classical paper [138] because it does not make use of any tools outside of the theory of Schur functions and Schur sequences. A closer look at the behaviour of distinguished subclasses of Sp×q(D) with respect to the SP-algorithm enables us to handle the corresponding versions of the matricial Schur problem restricted to these subclasses. Mathematics Subject Classification (2000). Primary 30E05, 47A57.Keywords. Schur function, matricial Schur problem, Schur-Potapov matrix polynomials, Arov-Kreȋn matrix polynomials. IntroductionIn the first part [57] of this paper we have established an appropriate adaptation of the Schur-Potapov algorithm for functions belonging to the matricial Schur class S p×q (D) to infinite sequences of complex p × q matrices. Using the fact that the Taylor coefficients sequences of functions from S p×q (D) are infinite p × q Schur sequences we mainly focused on the subclass S p×q,∞ (D) of S p×q (D) which consists of all p × q Schur functions for which the corresponding Taylor coefficient sequences are nondegenerate p × q Schur sequences. Thus, an one-to-one correspondence between infinite nondegenerate p × q Schur sequences and the set of all

show abstract

“…In [8] it is applied to solve the problem: When is a formal power series around z = 0 the Taylor expansion of a generalized Schur function. In [1,2,4,6,17] it is studied for its effect on the coisometric and unitary operator realizations of a generalized Schur function, including those whose state spaces are the reproducing kernel Pontryagin spaces with kernels K s (z, w) and D s (z, w); in [3] it is shown to provide an algorithm for the unique factorization of a 2 × 2 matrix polynomial which is J-unitary on T (for the definition, see below) in normalized elementary factors; and, finally, in [5] (see also [12]) it is used in solving a basic interpolation problem for generalized Schur functions.…”

Section: The Function S(z) Admits the Krein-langer Factorizationmentioning

confidence: 99%

Generalized Schur Functions and Augmented Schur Parameters

Dijksma

Wanjala

Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems

View full text Add to dashboard Cite

An augmented generalized happy function, S [c,b] maps a positive integer to the sum of the squares of its base-b digits and a non-negative integer c. A positive integer u is in a cycle of S [c,b] if, for some positive integer k, S k [c,b] (u) = u and for positive integers v and w, v is w-attracted for S [c,b] if, for some non-negative integer ℓ, S ℓ [c,b] (v) = w. In this paper, we prove that for each c ≥ 0 and b ≥ 2, and for any u in a cycle of S [c,b] , (1) if b is even, then there exist arbitrarily long sequences of consecutive u-attracted integers and (2) if b is odd, then there exist arbitrarily long sequences of 2-consecutive u-attracted integ 1. Introduction. Letting S 2 be the function that takes a positive integer to the sum of the squares of its (base 10) digits, a positive integer a is said to be a happy number if S k 2 (a) = 1 for some k ∈ Z + [5, 6]. These ideas were generalized in [2] as follows: Fix an integer b ≥ 2, and let a = n i=0 a i b i , where 0 ≤ a i ≤ b − 1 are integers. For each integer e ≥ 2, define the function S e,b : Z + → Z + by

show abstract

The Schur Algorithm for Generalized Schur Functions IV: Unitary Realizations

Cited by 11 publications

References 8 publications

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

Generalized Schur Functions and Augmented Schur Parameters

Contact Info

Product

Resources

About