van der Corput lemma
E776133
The van der Corput lemma is a fundamental result in analytic number theory and harmonic analysis that provides bounds for oscillatory integrals and exponential sums, crucial for estimating error terms in various asymptotic formulas.
All labels observed (2)
| Label | Occurrences |
|---|---|
| van der Corput lemma canonical | 1 |
| van der Corput lemma in harmonic analysis | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9070944 — 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: van der Corput lemma Context triple: [Johannes G. van der Corput, knownFor, van der Corput lemma]
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A.
Riemann–Lebesgue lemma
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.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
D.
Bombieri–Vinogradov theorem
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
E.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
- 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: van der Corput lemma Target entity description: The van der Corput lemma is a fundamental result in analytic number theory and harmonic analysis that provides bounds for oscillatory integrals and exponential sums, crucial for estimating error terms in various asymptotic formulas.
-
A.
Riemann–Lebesgue lemma
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.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
D.
Bombieri–Vinogradov theorem
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
E.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical lemma
ⓘ
result in harmonic analysis ⓘ |
| appearsIn |
monographs on exponential sums
ⓘ
texts on analytic number theory ⓘ texts on harmonic analysis ⓘ |
| appliesTo |
exponential sums
ⓘ
oscillatory integrals ⓘ trigonometric integrals ⓘ |
| assumes |
nonvanishing derivative or curvature conditions
ⓘ
smoothness conditions on the phase function ⓘ |
| field |
analytic number theory
ⓘ
harmonic analysis ⓘ |
| hasFormulation |
discrete exponential sum version
ⓘ
higher-derivative version for oscillatory integrals ⓘ one-dimensional oscillatory integral estimate ⓘ |
| hasVersion |
van der Corput A-process
NERFINISHED
ⓘ
van der Corput B-process NERFINISHED ⓘ |
| implies |
cancellation in exponential sums
ⓘ
decay of oscillatory integrals under derivative conditions ⓘ |
| influenced |
development of modern exponential sum techniques
ⓘ
methods for bounding oscillatory integrals in analysis ⓘ |
| namedAfter | J. G. van der Corput NERFINISHED ⓘ |
| provides |
bounds for exponential sums
ⓘ
bounds for oscillatory integrals ⓘ |
| relatedTo |
Weyl differencing
NERFINISHED
ⓘ
Weyl’s inequality NERFINISHED ⓘ equidistribution modulo 1 ⓘ method of exponential sums ⓘ stationary phase method ⓘ van der Corput inequality NERFINISHED ⓘ |
| timePeriod | 20th century mathematics ⓘ |
| typicalConclusion | integral bounded by constant times lambda to a negative power ⓘ |
| typicalHypothesis | k-th derivative of phase bounded away from zero ⓘ |
| usedFor |
bounding Weyl sums
ⓘ
bounding exponential sums over integers ⓘ bounding oscillatory integrals with phase functions ⓘ bounding trigonometric sums ⓘ estimating error terms in asymptotic formulas ⓘ proving equidistribution results ⓘ proving estimates in the circle method ⓘ |
| usedIn |
Fourier analysis on the real line
ⓘ
discrepancy theory ⓘ estimates for exponential integrals in PDE ⓘ metric number theory ⓘ proofs of bounds for the Riemann zeta function ⓘ |
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: van der Corput lemma Description of subject: The van der Corput lemma is a fundamental result in analytic number theory and harmonic analysis that provides bounds for oscillatory integrals and exponential sums, crucial for estimating error terms in various asymptotic formulas.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
van der Corput lemma in harmonic analysis