Triple

T8640756

Position	Surface form	Disambiguated ID	Type / Status
Subject	Frigyes Riesz	`E204639`	entity
Predicate	knownFor	`P22`	FINISHED
Object	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.	`E746579`	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 transforms | Statement: [Frigyes Riesz, knownFor, Riesz transforms]

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Riesz transforms
Context triple: [Frigyes Riesz, knownFor, Riesz transforms]

A. 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.
B. Three regularity results in harmonic analysis
"Three regularity results in harmonic analysis" is the doctoral thesis of mathematician Terence Tao, focusing on advanced problems in harmonic analysis and the study of regularity properties of functions and operators.
C. Calderón–Zygmund theory
Calderón–Zygmund theory is a branch of harmonic analysis that studies singular integral operators and their boundedness properties on function spaces such as L^p.
D. Singular Integrals and Differentiability Properties of Functions
"Singular Integrals and Differentiability Properties of Functions" is a landmark mathematical monograph by Elias M. Stein that developed the modern theory of singular integral operators and their role in harmonic analysis and differentiability.
E. Littlewood–Paley theory
Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
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 transforms
Triple: [Frigyes Riesz, knownFor, Riesz transforms]

Generated description

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.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Riesz transforms
Target entity description: 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.

A. 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.
B. Three regularity results in harmonic analysis
"Three regularity results in harmonic analysis" is the doctoral thesis of mathematician Terence Tao, focusing on advanced problems in harmonic analysis and the study of regularity properties of functions and operators.
C. Calderón–Zygmund theory
Calderón–Zygmund theory is a branch of harmonic analysis that studies singular integral operators and their boundedness properties on function spaces such as L^p.
D. Singular Integrals and Differentiability Properties of Functions
"Singular Integrals and Differentiability Properties of Functions" is a landmark mathematical monograph by Elias M. Stein that developed the modern theory of singular integral operators and their role in harmonic analysis and differentiability.
E. Littlewood–Paley theory
Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
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_69cebc340528819085c80b69d6d32a34`	completed	April 2, 2026, 6:57 p.m.
NEDg	Description generation	`batch_69cebcdb197881908eeb6c5e4b225a2c`	completed	April 2, 2026, 7 p.m.
NED2	Entity disambiguation (via description)	`batch_69cebd85ad8481909518afd02dbc5358`	completed	April 2, 2026, 7:03 p.m.

Created at: March 30, 2026, 6:28 p.m.