Multi-Point Degenerate Interpolation Problem for Generalized Schur Functions: Description of All Solutions

2010 ◽  
Vol 11 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Vladimir Bolotnikov
Keyword(s):  
Interpolation Problem ◽  
Schur Functions ◽  
Generalized Schur Functions ◽  
All Solutions

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

Analysis ◽  
2008 ◽  
Vol 28 (1) ◽  
Author(s):  
Adhemar Bultheel ◽  
Andreas Lasarow
Keyword(s):  
Unique Solution ◽  
Interpolation Problem ◽  
Schur Functions ◽  
Rational Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Orthogonal Rational Functions ◽  
All Solutions ◽  
Pick Problem ◽  
Complex Valued

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.

scholarly journals SCHUR–NEVANLINNA SEQUENCES OF RATIONAL FUNCTIONS

2007 ◽  
Vol 50 (3) ◽  
pp. 571-596 ◽  
Author(s):  
Adhemar Bultheel ◽  
Andreas Lasarow
Keyword(s):  
Unit Circle ◽  
Interpolation Problem ◽  
Schur Functions ◽  
Rational Functions ◽  
The Other ◽  
Complex Numbers ◽  
Orthogonal Rational Functions ◽  
All Solutions ◽  
The One

AbstractWe study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

A Basic Interpolation Problem for Generalized Schur Functions and Coisometric Realizations

2003 ◽  
pp. 39-76 ◽  
Author(s):  
D. Alpay ◽  
T. Ya. Azizov ◽  
A. Dijksma ◽  
H. Langer ◽  
G. Wanjala
Keyword(s):  
Interpolation Problem ◽  
Schur Functions ◽  
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.

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