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 (1)
| Label | Occurrences |
|---|---|
| Dirichlet theorem on Fourier series canonical | 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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| oftenStatedFor | functions of bounded variation ⓘ |
| relatedTo |
Dirichlet kernel
NERFINISHED
ⓘ
Gibbs phenomenon NERFINISHED ⓘ 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 NERFINISHED ⓘ |
| weakerThan |
Carleson theorem on almost everywhere convergence of Fourier series
NERFINISHED
ⓘ
results assuming higher smoothness such as C^1 or C^2 periodic functions ⓘ |
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.