Fourier restriction theory
E792937
Fourier restriction theory is a branch of harmonic analysis that studies when and how the Fourier transform of a function can be meaningfully restricted to lower-dimensional subsets such as curves or surfaces.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Fourier restriction norm method | 1 |
| Fourier restriction theory canonical | 1 |
| Stein restriction conjecture | 1 |
| Stein–Tomas restriction theorem | 1 |
| Tomas–Stein restriction theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9343779 — 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: Fourier restriction theory Context triple: [Lawrence Guth, researchInterest, Fourier restriction theory]
-
A.
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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B.
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.
-
C.
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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D.
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.
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E.
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.
- 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: Fourier restriction theory Target entity description: Fourier restriction theory is a branch of harmonic analysis that studies when and how the Fourier transform of a function can be meaningfully restricted to lower-dimensional subsets such as curves or surfaces.
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
branch of harmonic analysis
ⓘ
mathematical theory ⓘ |
| aimsTo |
characterize boundedness of restriction operators
ⓘ
obtain sharp Lp estimates for Fourier restriction ⓘ |
| concerns |
how the Fourier transform can be restricted to a subset
ⓘ
when the Fourier transform can be restricted to a subset ⓘ |
| context | Euclidean harmonic analysis ⓘ |
| developedBy |
Charles Fefferman
ⓘ
Elias M. Stein ⓘ
linked to:
Elias Stein
Jean Bourgain ⓘ Larry Guth ⓘ Peter Tomas ⓘ Terence Tao ⓘ Thomas Wolff ⓘ |
| field | harmonic analysis ⓘ |
| formalObject | restriction operator of the Fourier transform ⓘ |
| hasApplication |
Schrödinger equation
ⓘ
dispersive PDEs ⓘ nonlinear dispersive equations ⓘ wave equation ⓘ |
| hasProblem |
Stein restriction conjecture
ⓘ
linked to:
Fourier restriction theory
Tomas–Stein restriction theorem ⓘ
linked to:
Fourier restriction theory
restriction problem for the cone ⓘ restriction problem for the paraboloid ⓘ restriction problem for the sphere ⓘ |
| relatedTo |
Bochner–Riesz problem
ⓘ
linked to:
Bochner–Riesz means
Kakeya problem ⓘ Strichartz estimates ⓘ
linked to:
Strichartz-type estimates
decoupling theory ⓘ dispersive partial differential equations ⓘ local smoothing estimates ⓘ oscillatory integral operators ⓘ |
| studies |
Fourier transform on curves
ⓘ
Fourier transform on lower-dimensional subsets ⓘ Fourier transform on surfaces ⓘ restriction of the Fourier transform ⓘ |
| timePeriod | late 20th century ⓘ |
| typicalQuestion | for which exponents p and q does the restriction operator extend boundedly from Lp to Lq ⓘ |
| typicalSubset |
cone
ⓘ
curves in Euclidean space ⓘ hypersurfaces with nonvanishing curvature ⓘ paraboloid ⓘ sphere ⓘ |
| usesConcept |
Fourier transform
ⓘ
Lebesgue spaces ⓘ Littlewood–Paley theory ⓘ Lp spaces ⓘ
linked to:
Lebesgue spaces
decoupling inequalities ⓘ interpolation theory ⓘ oscillatory integrals ⓘ stationary phase method ⓘ wave packet decomposition ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Fourier restriction theory Description of subject: Fourier restriction theory is a branch of harmonic analysis that studies when and how the Fourier transform of a function can be meaningfully restricted to lower-dimensional subsets such as curves or surfaces.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Fourier restriction theory
linked to: Fourier restriction theory
linked to: Fourier restriction theory
linked to: Fourier restriction theory