The Schur algorithm for generalized Schur functions I: coisometric realizations

2001 ◽  
pp. 1-36 ◽  
Author(s):  
Daniel Alpay ◽  
Tomas Azizov ◽  
Aad Dijksma ◽  
Heinz Langer
Keyword(s):  
Schur Functions ◽  
Schur Algorithm ◽  
Generalized Schur Functions

scholarly journals BOUNDARY ANGULAR DERIVATIVES OF GENERALIZED SCHUR FUNCTIONS

2012 ◽  
Vol 93 (3) ◽  
pp. 203-224
Author(s):  
VLADIMIR BOLOTNIKOV ◽  
TENGYAO WANG ◽  
JOSHUA M. WEISS
Keyword(s):  
Schur Functions ◽  
Taylor Coefficients ◽  
Generalized Schur Functions ◽  
Angular Derivatives ◽  
Derivatives Of

AbstractCharacterization of generalized Schur functions in terms of their Taylor coefficients was established by Krein and Langer [‘Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume ${\Pi }_{\kappa } $ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen’, Math. Nachr. 77 (1977), 187–236]. We establish a boundary analogue of this characterization.

scholarly journals Generalized Schur–Nevanlinna functions and their realizations

2020 ◽  
Vol 92 (5) ◽  
Author(s):  
Lassi Lilleberg
Keyword(s):  
Transfer Functions ◽  
Schur Functions ◽  
Inner Function ◽  
Pontryagin Space ◽  
State Spaces ◽  
Limit Values ◽  
Adjoint Systems ◽  
Generalized Schur Functions ◽  
Nevanlinna Functions ◽  
Weak Similarity

Abstract Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which allows one to generalize the notion of an inner function to Pontryagin space operator valued setting. Transfer functions of self-adjoint systems such that their state spaces are Pontryagin spaces, are generalized Nevanlinna functions, and symmetric generalized Schur functions can be realized as transfer functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion when a symmetric generalized Schur function is also a generalized Nevanlinna function is given. The criterion involves the negative index of the weak similarity mapping between an optimal minimal realization and its dual. In the special case corresponding to the generalization of an inner function, a concrete model for the weak similarity mapping can be obtained by using the canonical realizations.

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