Dirichlet theorem on Fourier series
E898518
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Dirichlet theorem on Fourier series canonical | 1 |
| Dirichlet’s convergence theorem for Fourier series of piecewise smooth functions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992352 — 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: Dirichlet theorem on Fourier series Context triple: [Dirichlet conditions, relatedTo, Dirichlet theorem on Fourier series]
-
A.
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.
-
B.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
-
C.
Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale
"Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" is a classic mathematical treatise by Ulisse Dini on Fourier series and related analytic methods for representing real-valued functions.
-
D.
Lectures on Fourier Integrals
Lectures on Fourier Integrals is a classic mathematical monograph by Salomon Bochner that systematically develops the theory and applications of Fourier integrals and transforms.
-
E.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
- 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: Dirichlet theorem on Fourier series Target entity description: 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.
-
A.
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.
-
B.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
-
C.
Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale
"Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" is a classic mathematical treatise by Ulisse Dini on Fourier series and related analytic methods for representing real-valued functions.
-
D.
Lectures on Fourier Integrals
Lectures on Fourier Integrals is a classic mathematical monograph by Salomon Bochner that systematically develops the theory and applications of Fourier integrals and transforms.
-
E.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in Fourier analysis ⓘ |
| appliesTo |
2π-periodic functions
ⓘ
periodic functions of real variable ⓘ |
| assumes |
discontinuities are finite (no infinite jumps)
ⓘ
function has a finite number of discontinuities on a period ⓘ function has a finite number of maxima and minima on a period ⓘ function is 2π-periodic ⓘ |
| clarifies |
behavior of Fourier series at discontinuities
ⓘ
conditions for convergence of Fourier series ⓘ |
| concerns |
pointwise convergence of Fourier series
ⓘ
representation of periodic functions by trigonometric series ⓘ |
| concludes |
Fourier series converges at each point of continuity
ⓘ
Fourier series converges at each point of discontinuity to the midpoint of the left and right limits ⓘ Fourier series converges to (f(x+0)+f(x-0))/2 at jump discontinuities ⓘ Fourier series converges to f(x) where f is continuous ⓘ |
| doesNotGuarantee |
absolute convergence of Fourier series
ⓘ
uniform convergence ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ mathematical analysis ⓘ |
| givesConditionOn |
piecewise continuously differentiable functions
ⓘ
piecewise monotone functions ⓘ piecewise smooth functions ⓘ |
| hasApplicationIn |
heat equation
ⓘ
partial differential equations ⓘ signal processing ⓘ vibrating string problem ⓘ |
| historicalPeriod | 19th century ⓘ |
| implies |
Fourier series of a function of bounded variation converges at every point
ⓘ
Fourier series of a function of bounded variation converges to the average of one-sided limits at each point ⓘ Fourier series of a piecewise smooth periodic function converges everywhere ⓘ Fourier series represents the function almost everywhere under its hypotheses ⓘ |
| isTaughtIn |
Fourier analysis courses
ⓘ
advanced calculus courses ⓘ real analysis courses ⓘ |
| namedAfter |
Johann Peter Gustav Lejeune Dirichlet
ⓘ
linked to:
Peter Gustav Lejeune Dirichlet
|
| oftenStatedFor | functions of bounded variation ⓘ |
| relatedTo |
Dirichlet kernel
ⓘ
Gibbs phenomenon ⓘ pointwise convergence of trigonometric series ⓘ uniform convergence of Fourier series ⓘ |
| strongerThan | basic convergence results for continuous periodic functions with piecewise continuous derivative ⓘ |
| typeOfConvergence | pointwise convergence ⓘ |
| uses | Dirichlet kernel ⓘ |
| weakerThan |
Carleson theorem on almost everywhere convergence of Fourier series
ⓘ
results assuming higher smoothness such as C^1 or C^2 periodic functions ⓘ |
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: Dirichlet theorem on Fourier series Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Dirichlet kernel
→
usedInProofOf
→
Dirichlet’s convergence theorem for Fourier series of piecewise smooth functions
ⓘ
linked to: Dirichlet theorem on Fourier series