Triple

T5709654
Position Surface form Disambiguated ID Type / Status
Subject Elias Stein E125871 entity
Predicate notableWork P4 FINISHED
Object Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
"Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" is a foundational graduate-level textbook by Elias Stein that systematically develops modern harmonic analysis using real-variable techniques, emphasizing singular integrals, Littlewood–Paley theory, and oscillatory integral methods.
E542636 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: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals | Statement: [Elias Stein, notableWork, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
Context triple: [Elias Stein, notableWork, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals]
  • A. 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.
  • B. 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.
  • C. 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.
  • D. Hardy–Littlewood maximal function
    The Hardy–Littlewood maximal function is a fundamental operator in real analysis and harmonic analysis that controls the local averages of a function and plays a key role in differentiation theorems and singular integral theory.
  • E. Fefferman–Phong inequality
    The Fefferman–Phong inequality is a fundamental result in harmonic analysis and partial differential equations that provides weighted \(L^2\) estimates controlling functions by their gradients and associated potentials.
  • 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: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
Triple: [Elias Stein, notableWork, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals]
Generated description
"Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" is a foundational graduate-level textbook by Elias Stein that systematically develops modern harmonic analysis using real-variable techniques, emphasizing singular integrals, Littlewood–Paley theory, and oscillatory integral methods.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
Target entity description: "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" is a foundational graduate-level textbook by Elias Stein that systematically develops modern harmonic analysis using real-variable techniques, emphasizing singular integrals, Littlewood–Paley theory, and oscillatory integral methods.
  • A. 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.
  • B. 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.
  • C. 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.
  • D. Hardy–Littlewood maximal function
    The Hardy–Littlewood maximal function is a fundamental operator in real analysis and harmonic analysis that controls the local averages of a function and plays a key role in differentiation theorems and singular integral theory.
  • E. Fefferman–Phong inequality
    The Fefferman–Phong inequality is a fundamental result in harmonic analysis and partial differential equations that provides weighted \(L^2\) estimates controlling functions by their gradients and associated potentials.
  • 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_69c0082d6fe48190b777fb383769e5c8 completed March 22, 2026, 3:18 p.m.
NER Named-entity recognition batch_69c0248c3dac8190824fca9ddde89665 completed March 22, 2026, 5:19 p.m.
NED1 Entity disambiguation (via context triple) batch_69c05a6f5ac08190b5acbccea756d2de completed March 22, 2026, 9:09 p.m.
NEDg Description generation batch_69c062029e3c8190ade3f0836d6b3842 completed March 22, 2026, 9:41 p.m.
NED2 Entity disambiguation (via description) batch_69c06268eb0c8190959ba762c2d9b47d completed March 22, 2026, 9:43 p.m.
Created at: March 22, 2026, 3:46 p.m.