Riemann–Lebesgue lemma
E47351
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Riemann–Lebesgue lemma canonical | 4 |
| Riemann–Lebesgue lemma on locally compact abelian groups | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T373783 — 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: Riemann–Lebesgue lemma Context triple: [Bernhard Riemann, knownFor, Riemann–Lebesgue lemma]
-
A.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
B.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- 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: Riemann–Lebesgue lemma Target entity description: The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
A.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
B.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
lemma
ⓘ
mathematical theorem ⓘ |
| appearsIn |
textbooks on Lebesgue integration
ⓘ
textbooks on harmonic analysis ⓘ |
| appliesTo |
Fourier analysis
ⓘ
surface form:
Fourier series
Fourier analysis ⓘ
surface form:
Fourier transform
L¹ functions ⓘ integrable functions ⓘ |
| assumption |
function belongs to L¹ with respect to the relevant measure
ⓘ
function is integrable in the Lebesgue sense ⓘ |
| conclusion |
Fourier coefficients of an L¹ function vanish at infinity.
ⓘ
Fourier transform of an L¹ function vanishes at infinity. ⓘ |
| contrastWith |
pointwise convergence of Fourier series
ⓘ
uniform convergence of Fourier series ⓘ |
| doesNotRequire |
function to be continuous
ⓘ
function to be square-integrable ⓘ |
| domain |
2π-periodic integrable functions
ⓘ
functions on the real line ⓘ functions on ℝⁿ ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ real analysis ⓘ |
| generalization |
Riemann–Lebesgue lemma
self-linksurface differs
ⓘ
surface form:
Riemann–Lebesgue lemma on locally compact abelian groups
|
| holdsFor |
absolutely integrable functions
ⓘ
compactly supported integrable functions ⓘ |
| implies |
Fourier coefficients of an L¹ function form a null sequence.
ⓘ
Fourier transform of an L¹ function is a continuous function that tends to 0 at infinity. ⓘ |
| isWeakerThan | results that give rates of decay of Fourier coefficients ⓘ |
| namedAfter |
Bernhard Riemann
ⓘ
Henri Lebesgue ⓘ |
| relatedTo |
Fourier coefficient
ⓘ
Fourier inversion theorem ⓘ Fourier analysis ⓘ
surface form:
Fourier transform
Lebesgue integration ⓘ
surface form:
Lebesgue integral
L¹ space ⓘ Plancherel theorem for real reductive groups ⓘ
surface form:
Plancherel theorem
trigonometric series ⓘ |
| statement |
If f is in L¹(ℝⁿ), then its Fourier transform tends to 0 at infinity.
ⓘ
If f is integrable on [−π,π], then its Fourier coefficients tend to 0 as the frequency index tends to infinity. ⓘ |
| topicIn |
graduate Fourier analysis courses
ⓘ
graduate real analysis courses ⓘ |
| typeOfLimit | asymptotic vanishing of frequency components ⓘ |
| usedIn |
approximation theory
ⓘ
harmonic analysis on locally compact abelian groups ⓘ proofs of convergence results for Fourier series ⓘ signal processing theory ⓘ |
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: Riemann–Lebesgue lemma Description of subject: The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Riemann–Lebesgue lemma on locally compact abelian groups
subject surface form:
Georg Friedrich Bernhard Riemann
subject surface form:
Friedrich Bernhard Riemann