Dini test for convergence of Fourier series
E877690
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Dini test for convergence of Fourier series canonical | 1 |
| Dini–Lipschitz test | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10660072 — 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: Dini test for convergence of Fourier series Context triple: [Ulisse Dini, notableConcept, Dini test for convergence of Fourier series]
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A.
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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B.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
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C.
Dirichlet test
The Dirichlet test is a criterion in mathematical analysis that provides sufficient conditions for the convergence of certain infinite series, particularly those involving oscillatory terms.
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D.
Fourier series
A Fourier series is a way of representing a periodic function as an infinite sum of sines and cosines with appropriately chosen coefficients.
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E.
The Fourier Integral and Certain of Its Applications
The Fourier Integral and Certain of Its Applications is a foundational mathematical work by Norbert Wiener that develops and applies Fourier analysis to problems in harmonic analysis and related areas.
- 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: Dini test for convergence of Fourier series Target entity description: 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.
-
A.
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.
-
B.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
-
C.
Dirichlet test
The Dirichlet test is a criterion in mathematical analysis that provides sufficient conditions for the convergence of certain infinite series, particularly those involving oscillatory terms.
-
D.
Fourier series
A Fourier series is a way of representing a periodic function as an infinite sum of sines and cosines with appropriately chosen coefficients.
-
E.
The Fourier Integral and Certain of Its Applications
The Fourier Integral and Certain of Its Applications is a foundational mathematical work by Norbert Wiener that develops and applies Fourier analysis to problems in harmonic analysis and related areas.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
convergence test
ⓘ
criterion for pointwise convergence ⓘ theorem in harmonic analysis ⓘ |
| appearsIn | classical textbooks on Fourier analysis ⓘ |
| appliesTo |
Fourier series
ⓘ
trigonometric Fourier series ⓘ |
| assumptionOnPoint | x₀ is a Lebesgue point under the Dini condition ⓘ |
| concerns |
behavior of a function near a point
ⓘ
pointwise convergence of Fourier series ⓘ |
| conclusion |
Fourier series converges at x₀
ⓘ
limit equals (f(x₀+0)+f(x₀−0))/2 ⓘ |
| conditionType | sufficient but not necessary ⓘ |
| contrastsWith | global convergence criteria for Fourier series ⓘ |
| domain |
2π-periodic functions
ⓘ
integrable functions on an interval ⓘ |
| ensures |
convergence of the Fourier series to the average of one-sided limits
ⓘ
summability of Fourier series at a point under Dini condition ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ |
| focusesOn | local behavior of f near the convergence point ⓘ |
| generalizes | Jordan criterion for piecewise monotone functions ⓘ |
| gives | sufficient conditions for convergence ⓘ |
| guarantees | convergence to midpoint of jump at a Dini point ⓘ |
| historicalPeriod | late 19th century ⓘ |
| involves |
control of oscillation of f near the point
ⓘ
singular integral estimates ⓘ |
| isPartOf | local convergence theory of Fourier series ⓘ |
| languageOfFormulation | real analysis ⓘ |
| mathematicalSubjectClassification | 42A20 ⓘ |
| namedAfter | Ulisse Dini ⓘ |
| relatedTo |
Dini–Lipschitz test
ⓘ
linked to:
Dini test for convergence of Fourier series
Dirichlet test for Fourier series ⓘ
linked to:
Dirichlet test
Jordan test for convergence of Fourier series ⓘ |
| requires |
existence of one-sided limits at the point
ⓘ
integral condition on the modulus of continuity near the point ⓘ |
| strength |
stronger than mere existence of one-sided limits
ⓘ
weaker than uniform continuity assumptions ⓘ |
| typicalAssumptionOnFunction | f is 2π-periodic and integrable on [-π,π] ⓘ |
| typicalCondition |
integral from 0 to δ of |f(x₀+t)+f(x₀−t)−2s|/t dt is finite for some δ>0
ⓘ
integral from 0 to δ of ω(f;x₀,t)/t dt is finite, where ω is a local modulus of continuity ⓘ |
| typicalFunctionSpace | L¹([-π,π]) ⓘ |
| typicalPointNotation | x₀ ⓘ |
| typicalUseCase | functions with jump discontinuities ⓘ |
| usedIn |
analysis of Gibbs phenomenon near discontinuities
ⓘ
classical theory of Fourier series ⓘ study of boundary behavior of harmonic functions ⓘ |
| uses | Dirichlet kernel ⓘ |
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Input
Subject: Dini test for convergence of Fourier series Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Dini test for convergence of Fourier series