Triple

T8640758
Position Surface form Disambiguated ID Type / Status
Subject Frigyes Riesz E204639 entity
Predicate knownFor P22 FINISHED
Object Riesz projection
The Riesz projection is a linear operator in functional analysis that projects onto the invariant subspace associated with a portion of the spectrum of a bounded linear operator, defined via contour integration of its resolvent.
E747348 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Riesz projection | Statement: [Frigyes Riesz, knownFor, Riesz projection]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Riesz projection
Context triple: [Frigyes Riesz, knownFor, Riesz projection]
  • A. Runge approximation theorem
    The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
  • B. Bochner–Riesz means
    Bochner–Riesz means are a family of summability methods in harmonic analysis used to improve the convergence of Fourier series and Fourier integrals by smoothing their partial sums.
  • C. Riesz representation theorem
    The Riesz representation theorem is a fundamental result in functional analysis that characterizes continuous linear functionals on Hilbert spaces as inner products with a unique vector in the space.
  • D. 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.
  • E. 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.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Riesz projection
Triple: [Frigyes Riesz, knownFor, Riesz projection]
Generated description
The Riesz projection is a linear operator in functional analysis that projects onto the invariant subspace associated with a portion of the spectrum of a bounded linear operator, defined via contour integration of its resolvent.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Riesz projection
Target entity description: The Riesz projection is a linear operator in functional analysis that projects onto the invariant subspace associated with a portion of the spectrum of a bounded linear operator, defined via contour integration of its resolvent.
  • A. Riesz transforms
    Riesz transforms are fundamental singular integral operators in harmonic analysis that generalize the Hilbert transform to higher dimensions and play a key role in studying function spaces and partial differential equations.
  • B. Riesz–Thorin interpolation theorem
    The Riesz–Thorin interpolation theorem is a fundamental result in functional analysis that provides bounds for linear operators between Lᵖ spaces by interpolating their behavior between two known endpoint estimates.
  • C. Riesz
    Riesz is a Hungarian surname most notably associated with the influential mathematician Frigyes Riesz, a pioneer in functional analysis.
  • D. Runge approximation theorem
    The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
  • E. Riesz lemma
    Riesz lemma is a fundamental result in functional analysis that characterizes how, in an infinite-dimensional normed space, one can find unit vectors that stay a fixed distance away from any given proper closed subspace.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69ca834ca1c88190a11ffb0200342fac completed March 30, 2026, 2:06 p.m.
NER Named-entity recognition batch_69cc47944d1c819081f448f14d04bf9d completed March 31, 2026, 10:15 p.m.
NED1 Entity disambiguation (via context triple) batch_69cebc3b1f508190978df29d995f494c completed April 2, 2026, 6:58 p.m.
NEDg Description generation batch_69cebf12eec081909c40ccced9c1c52b completed April 2, 2026, 7:10 p.m.
NED2 Entity disambiguation (via description) batch_69cebf7302b08190bdfc84aac3147954 completed April 2, 2026, 7:11 p.m.
Created at: March 30, 2026, 6:28 p.m.