Schur algorithm
E506855
The Schur algorithm is a recursive procedure in complex analysis and operator theory used to construct and analyze Schur functions, playing a key role in interpolation problems and system theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Schur algorithm canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5256424 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Schur algorithm Context triple: [Carathéodory–Fejér interpolation, relatedTo, Schur algorithm]
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A.
Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
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B.
Szegő limit theorem
The Szegő limit theorem is a fundamental result in analysis and operator theory that describes the asymptotic behavior of determinants of large Toeplitz matrices in terms of the symbol’s integral.
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C.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
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D.
Hermite–Biehler theorem
The Hermite–Biehler theorem is a result in complex analysis and control theory that characterizes when a complex polynomial has all its zeros in the open upper half-plane in terms of the interlacing of zeros of two associated real polynomials.
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E.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Schur algorithm Target entity description: The Schur algorithm is a recursive procedure in complex analysis and operator theory used to construct and analyze Schur functions, playing a key role in interpolation problems and system theory.
-
A.
Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
-
B.
Szegő limit theorem
The Szegő limit theorem is a fundamental result in analysis and operator theory that describes the asymptotic behavior of determinants of large Toeplitz matrices in terms of the symbol’s integral.
-
C.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
-
D.
Hermite–Biehler theorem
The Hermite–Biehler theorem is a result in complex analysis and control theory that characterizes when a complex polynomial has all its zeros in the open upper half-plane in terms of the interlacing of zeros of two associated real polynomials.
-
E.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical algorithm
ⓘ
method in complex analysis ⓘ method in operator theory ⓘ recursive procedure ⓘ |
| appliesTo | Schur functions ⓘ |
| assumes |
function analytic in the open unit disk
ⓘ
function bounded by 1 in modulus on the unit disk ⓘ |
| basedOn | Schur transformation NERFINISHED ⓘ |
| characterizes | contractive analytic functions on the unit disk ⓘ |
| domain | unit disk ⓘ |
| field |
complex analysis
ⓘ
control theory ⓘ function theory ⓘ operator theory ⓘ system theory ⓘ |
| generalizationOf | continued fraction expansions for analytic functions ⓘ |
| historicalPeriod | early 20th century ⓘ |
| input | Schur function ⓘ |
| mapsTo | unit ball of H-infinity ⓘ |
| namedAfter | Issai Schur NERFINISHED ⓘ |
| output |
sequence of Schur parameters
ⓘ
sequence of contractive coefficients ⓘ |
| property |
iteratively reduces degree or complexity of a Schur function
ⓘ
preserves contractivity at each step ⓘ |
| relatedTo |
Hardy spaces
NERFINISHED
ⓘ
Herglotz functions NERFINISHED ⓘ Nevanlinna–Pick interpolation problem NERFINISHED ⓘ Schur complement NERFINISHED ⓘ inner–outer factorization ⓘ transfer functions of linear systems ⓘ |
| usedFor |
Carathéodory–Fejér interpolation
NERFINISHED
ⓘ
Nevanlinna–Pick interpolation NERFINISHED ⓘ analysis of Schur functions ⓘ computation of Schur parameters ⓘ computation of reflection coefficients ⓘ construction of Schur functions ⓘ factorization of analytic functions ⓘ interpolation problems ⓘ model reduction in system theory ⓘ realization theory in system theory ⓘ signal processing ⓘ spectral estimation ⓘ |
| usedIn |
discrete-time system theory
ⓘ
operator model theory ⓘ orthogonal polynomials on the unit circle ⓘ prediction theory of stationary processes ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Schur algorithm Description of subject: The Schur algorithm is a recursive procedure in complex analysis and operator theory used to construct and analyze Schur functions, playing a key role in interpolation problems and system theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.