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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Carleson theorem | 1 |
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 ⓘ |
Referenced by (2)
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