Carleson theorem on almost-everywhere convergence
E898520
The Carleson theorem on almost-everywhere convergence is a fundamental result in harmonic analysis stating that the Fourier series of any square-integrable function on the circle converges almost everywhere to the function itself.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Carleson theorem | 1 |
| Carleson theorem on almost-everywhere convergence canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992355 — 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: Carleson theorem on almost-everywhere convergence Context triple: [Dirichlet conditions, contrastWith, Carleson theorem on almost-everywhere convergence]
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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.
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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.
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C.
Bochner–Riesz means
Bochner–Riesz means are a family of summability methods in harmonic analysis used to improve the convergence of Fourier series and Fourier integrals by smoothing their partial sums.
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D.
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.
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E.
Harmonic Analysis and the Theory of Probability
Harmonic Analysis and the Theory of Probability is a seminal mathematical monograph that connects Fourier-analytic methods with probabilistic concepts, helping to lay the foundations of modern probability theory.
- 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: Carleson theorem on almost-everywhere convergence Target entity description: The Carleson theorem on almost-everywhere convergence is a fundamental result in harmonic analysis stating that the Fourier series of any square-integrable function on the circle converges almost everywhere to the function itself.
-
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.
Bochner–Riesz means
Bochner–Riesz means are a family of summability methods in harmonic analysis used to improve the convergence of Fourier series and Fourier integrals by smoothing their partial sums.
-
D.
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.
-
E.
Harmonic Analysis and the Theory of Probability
Harmonic Analysis and the Theory of Probability is a seminal mathematical monograph that connects Fourier-analytic methods with probabilistic concepts, helping to lay the foundations of modern probability theory.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in harmonic analysis ⓘ |
| addresses | Lusin problem on pointwise convergence of Fourier series NERFINISHED ⓘ |
| appliesTo |
L^2 functions on the circle
ⓘ
square-integrable functions ⓘ |
| assumes | function is in L^2 of the circle ⓘ |
| category |
theorem about trigonometric series
ⓘ
theorem in real analysis ⓘ |
| concerns |
Fourier series
NERFINISHED
ⓘ
almost-everywhere convergence ⓘ |
| conclusion | partial sums of the Fourier series converge to the function almost everywhere ⓘ |
| connectedTo | maximal Fourier partial sum operator boundedness on L^2 ⓘ |
| doesNotRequire |
absolute convergence of Fourier series
ⓘ
function to be continuous ⓘ |
| domain |
one-dimensional torus
ⓘ
unit circle ⓘ |
| equivalentTo | L^2 boundedness of the Carleson maximal operator ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ |
| generalizationOf | results on convergence of Fourier series for continuous functions ⓘ |
| generalizedBy | Carleson–Hunt theorem NERFINISHED ⓘ |
| hasConsequence | Fourier series of L^2 functions represent the function almost everywhere ⓘ |
| hasImpactOn |
Fourier series theory on the torus
NERFINISHED
ⓘ
ergodic theory via analogous convergence questions ⓘ |
| historicalContext | settled a question going back to Lebesgue and others ⓘ |
| implies | pointwise convergence almost everywhere of Fourier series for L^2 functions ⓘ |
| influenced |
research on pointwise convergence of Fourier integrals
ⓘ
study of singular integral operators ⓘ |
| inspired | development of modern time-frequency analysis ⓘ |
| isNontrivialFor | functions not in L^p for p>2 ⓘ |
| namedAfter | Lennart Carleson NERFINISHED ⓘ |
| provedBy | Lennart Carleson NERFINISHED ⓘ |
| relatedOpenProblem | a.e. convergence of Fourier series for L^p with 1<p<2 before Hunt ⓘ |
| relatedTo |
Carleson operator
NERFINISHED
ⓘ
Carleson–Hunt theorem NERFINISHED ⓘ maximal partial sum operator for Fourier series ⓘ |
| resolves | longstanding open problem on a.e. convergence of Fourier series for L^2 functions ⓘ |
| space | L^2(T) ⓘ |
| statement | the Fourier series of any L^2 function on the circle converges almost everywhere to the function itself ⓘ |
| strengthens | convergence of Fourier series in L^2 norm ⓘ |
| technique |
phase-plane analysis
ⓘ
stopping-time arguments ⓘ |
| typeOfConvergence | pointwise almost-everywhere convergence ⓘ |
| uses | time-frequency analysis techniques ⓘ |
| yearProved | 1966 ⓘ |
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Input
Subject: Carleson theorem on almost-everywhere convergence Description of subject: The Carleson theorem on almost-everywhere convergence is a fundamental result in harmonic analysis stating that the Fourier series of any square-integrable function on the circle converges almost everywhere to the function itself.
Referenced by (2)
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