Carleson theorem on almost-everywhere convergence

E898520
mathematical theorem result in harmonic analysis

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.

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All labels observed (5)

Label Occurrences
Carleson–Hunt theorem 3
Carleson theorem on almost everywhere convergence of Fourier series 2
Carleson theorem 1

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Statements (45)

Predicate Object
instanceOf mathematical theorem ⓘ
result in harmonic analysis ⓘ
addresses Lusin problem on pointwise convergence of Fourier series ⓘ
linked to: Carleson theorem on almost-everywhere convergence
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 ⓘ
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 ⓘ
linked to: Carleson theorem on almost-everywhere convergence
hasConsequence Fourier series of L^2 functions represent the function almost everywhere ⓘ
hasImpactOn Fourier series theory on the torus ⓘ
linked to: Fourier series on the torus

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 ⓘ
provedBy Lennart Carleson ⓘ
relatedOpenProblem a.e. convergence of Fourier series for L^p with 1<p<2 before Hunt ⓘ
relatedTo Carleson operator ⓘ
Carleson–Hunt theorem ⓘ
linked to: Carleson theorem on almost-everywhere convergence

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 ⓘ

How these facts were elicited

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Dirichlet conditions → contrastWith → Carleson theorem on almost-everywhere convergence ⓘ
Lennart Carleson → knownFor → Carleson theorem ⓘ
subject linked to: Carleson
linked to: Carleson theorem on almost-everywhere convergence
Dirichlet theorem on Fourier series → weakerThan → Carleson theorem on almost everywhere convergence of Fourier series ⓘ
linked to: Carleson theorem on almost-everywhere convergence
Carleson theorem on almost-everywhere convergence → relatedTo → Carleson–Hunt theorem ⓘ
linked to: Carleson theorem on almost-everywhere convergence
Carleson theorem on almost-everywhere convergence → generalizedBy → Carleson–Hunt theorem ⓘ
linked to: Carleson theorem on almost-everywhere convergence
Carleson theorem on almost-everywhere convergence → addresses → Lusin problem on pointwise convergence of Fourier series ⓘ
linked to: Carleson theorem on almost-everywhere convergence
Lennart Carleson → knownFor → Carleson theorem on almost everywhere convergence of Fourier series ⓘ
linked to: Carleson theorem on almost-everywhere convergence
Lennart Carleson → knownFor → Carleson–Hunt theorem ⓘ
linked to: Carleson theorem on almost-everywhere convergence