Triple

T19456673
Position Surface form Disambiguated ID Type / Status
Subject Almost Periodic Functions E486749 entity
Predicate centralConcept P533 FINISHED
Object Bohr–Fourier series
The Bohr–Fourier series is a generalization of the classical Fourier series that represents almost periodic functions as infinite sums of complex exponentials with possibly incommensurable frequencies.
E1376596 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: Bohr–Fourier series | Statement: [Almost Periodic Functions, centralConcept, Bohr–Fourier series]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Bohr–Fourier series
Context triple: [Almost Periodic Functions, centralConcept, Bohr–Fourier series]
  • A. Fejér’s theorem on Fourier series
    Fejér’s theorem on Fourier series is a fundamental result in harmonic analysis stating that the Cesàro means (Fejér means) of the Fourier series of a continuous periodic function always converge uniformly to the function itself.
  • B. Dirichlet theorem on Fourier series
    The Dirichlet theorem on Fourier series gives conditions under which a periodic function can be represented by a convergent Fourier series, specifying how and where the series converges to the function.
  • C. Dini test for convergence of Fourier series
    The Dini test for convergence of Fourier series is a classical criterion in harmonic analysis that gives sufficient conditions, involving the behavior of a function near a point, to ensure the pointwise convergence of its Fourier series there.
  • D. Trigonometric Series, Vol. I
    Trigonometric Series, Vol. I is a foundational mathematical monograph by Antoni Zygmund that systematically develops the theory of trigonometric series and Fourier analysis.
  • E. Trigonometric Series, Vol. II
    "Trigonometric Series, Vol. II" is a classic advanced mathematics text by Antoni Zygmund that develops the modern theory of trigonometric series and Fourier analysis.
  • 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: Bohr–Fourier series
Triple: [Almost Periodic Functions, centralConcept, Bohr–Fourier series]
Generated description
The Bohr–Fourier series is a generalization of the classical Fourier series that represents almost periodic functions as infinite sums of complex exponentials with possibly incommensurable frequencies.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Bohr–Fourier series
Target entity description: The Bohr–Fourier series is a generalization of the classical Fourier series that represents almost periodic functions as infinite sums of complex exponentials with possibly incommensurable frequencies.
  • A. Fejér’s theorem on Fourier series
    Fejér’s theorem on Fourier series is a fundamental result in harmonic analysis stating that the Cesàro means (Fejér means) of the Fourier series of a continuous periodic function always converge uniformly to the function itself.
  • B. Dirichlet theorem on Fourier series
    The Dirichlet theorem on Fourier series gives conditions under which a periodic function can be represented by a convergent Fourier series, specifying how and where the series converges to the function.
  • C. Dini test for convergence of Fourier series
    The Dini test for convergence of Fourier series is a classical criterion in harmonic analysis that gives sufficient conditions, involving the behavior of a function near a point, to ensure the pointwise convergence of its Fourier series there.
  • D. Trigonometric Series, Vol. I
    Trigonometric Series, Vol. I is a foundational mathematical monograph by Antoni Zygmund that systematically develops the theory of trigonometric series and Fourier analysis.
  • E. Trigonometric Series, Vol. II
    "Trigonometric Series, Vol. II" is a classic advanced mathematics text by Antoni Zygmund that develops the modern theory of trigonometric series and Fourier analysis.
  • 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_69d8e8d86d608190bd199a98d0297f27 completed April 10, 2026, 12:11 p.m.
NER Named-entity recognition batch_69e633c4088881908f23f25a82a513f6 completed April 20, 2026, 2:10 p.m.
NED1 Entity disambiguation (via context triple) batch_6a073b386590819087632ce0b7375c58 completed May 15, 2026, 3:26 p.m.
NEDg Description generation batch_6a073c61707081909c25daa85176cdf4 completed May 15, 2026, 3:31 p.m.
NED2 Entity disambiguation (via description) batch_6a073cd498788190bc1d725aeff54c07 completed May 15, 2026, 3:33 p.m.
Created at: April 10, 2026, 1:38 p.m.