The Schur algorithm for generalized Schur functions I: coisometric realizations

Monatshefte f�r Mathematik

et al. 2003

Abstract. In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1-36) it was shown that for a generalized Schur function sðzÞ, which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal sðzÞ À1 of the characteristic function sðzÞ and general factorization results for characteristic functions.2000 Mathematics Subject Classification: 46C20, 47A48, 30D99

Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 84%

Section: Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions

Alpay

Azizov

Monatshefte f�r Mathematik

et al. 2003

Generalized Schur Functions and Augmented Schur Parameters

Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems

Wanjala

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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

Basic Boundary Interpolation for Generalized Schur Functions and Factorization of Rational J-unitary Matrix Functions

Alpay

Operator Theory: Advances and Applications

Langer

et al.

Self Cite

We define and solve a boundary interpolation problem for generalized Schur functions s(z) on the open unit disk D which have preassigned asymptotics when z from D tends nontangentially to a boundary point z1 ∈ T. The solutions are characterized via a fractional linear parametrization formula. We also prove that a rational J-unitary 2 × 2-matrix function whose only pole is at z1 has a unique minimal factorization into elementary factors and we classify these factors. The parametrization formula is then used in an algorithm for obtaining this factorization. In the proofs we use reproducing kernel space methods.