Bochner–Riesz means
E613406
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Bochner–Riesz means canonical | 1 |
| Riesz means | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6716264 — 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: Bochner–Riesz means Context triple: [Salomon Bochner, notableFor, Bochner–Riesz means]
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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.
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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C.
Littlewood–Paley theory
Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
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D.
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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E.
Calderón–Zygmund theory
Calderón–Zygmund theory is a branch of harmonic analysis that studies singular integral operators and their boundedness properties on function spaces such as L^p.
- 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: Bochner–Riesz means Target entity description: 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.
-
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.
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.
-
C.
Littlewood–Paley theory
Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
-
D.
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.
-
E.
Calderón–Zygmund theory
Calderón–Zygmund theory is a branch of harmonic analysis that studies singular integral operators and their boundedness properties on function spaces such as L^p.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
Fourier analysis technique
ⓘ
concept in harmonic analysis ⓘ summability method ⓘ |
| appearsIn |
dispersive estimates
ⓘ
partial differential equations ⓘ study of convergence of multiple Fourier series ⓘ |
| appliesTo |
Fourier integrals on Euclidean space
ⓘ
Fourier series on the torus NERFINISHED ⓘ |
| associatedWith | Bochner–Riesz conjecture NERFINISHED ⓘ |
| basedOn | Bochner–Riesz multipliers NERFINISHED ⓘ |
| contrastedWith | sharp spectral cutoff of partial sums ⓘ |
| definedOn | Fourier transform side ⓘ |
| dependsOn | radius parameter in frequency space ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ |
| generalizes | Riesz means NERFINISHED ⓘ |
| hasEffect | reducing Gibbs-type oscillations in some settings ⓘ |
| hasKernel | Bochner–Riesz kernel in physical space ⓘ |
| hasOpenProblem | Bochner–Riesz conjecture in higher dimensions NERFINISHED ⓘ |
| hasParameter |
dimension n
ⓘ
order parameter δ ⓘ |
| hasProperty | rotationally symmetric in frequency space ⓘ |
| improves |
norm convergence of Fourier series in some ranges of p and δ
ⓘ
pointwise convergence of Fourier series in some ranges of p and δ ⓘ |
| namedAfter |
Marcel Riesz
NERFINISHED
ⓘ
Salomon Bochner NERFINISHED ⓘ |
| relatedTo |
Cesàro means
ⓘ
Fejér means NERFINISHED ⓘ Littlewood–Paley theory NERFINISHED ⓘ Stein–Tomas restriction theorem NERFINISHED ⓘ maximal Bochner–Riesz operator ⓘ spherical summation methods ⓘ square function estimates ⓘ |
| studiedInContextOf |
Lp boundedness of Fourier multipliers
ⓘ
restriction theory for the Fourier transform ⓘ singular integrals ⓘ |
| typicalDomain |
Rn
ⓘ
n-dimensional torus ⓘ |
| usedBy |
analysts studying Fourier integrals
ⓘ
analysts studying Fourier series ⓘ |
| usedFor |
improving convergence of Fourier integrals
ⓘ
improving convergence of Fourier series ⓘ smoothing partial sums of Fourier integrals ⓘ smoothing partial sums of Fourier series ⓘ |
| uses | radial cutoff in frequency domain ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Bochner–Riesz means Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Riesz means