Kronecker’s lemma
E100233
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kronecker’s lemma canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T846897 — 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: Kronecker’s lemma Context triple: [Leopold Kronecker, notableWork, Kronecker’s 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.
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B.
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.
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C.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
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D.
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.
-
E.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
- 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: Kronecker’s lemma Target entity description: 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.
-
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.
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.
-
C.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
-
D.
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.
-
E.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical lemma
ⓘ
result in real analysis ⓘ result in summability theory ⓘ |
| appliesTo |
convergent series
ⓘ
weighted sums of partial sums ⓘ |
| assumes |
a_n is monotone decreasing to 0
ⓘ
partial sums s_n = x_1 + … + x_n ⓘ sequence of real numbers (a_n) ⓘ sequence of real numbers (x_n) ⓘ sum_{n=1}^∞ x_n converges ⓘ |
| concludes |
(1/A_n) * sum_{k=1}^n a_k s_k → 0 under suitable conditions
ⓘ
certain weighted averages of partial sums converge to 0 ⓘ |
| conclusionType | limit statement ⓘ |
| field |
Fourier analysis
ⓘ
ergodic theory ⓘ real analysis ⓘ summability theory ⓘ |
| hasVersion |
complex-valued sequence version
ⓘ
real-valued sequence version ⓘ |
| implies | Abel-type summability results in some contexts ⓘ |
| logicalForm | if-then statement about limits of sequences ⓘ |
| mathematicalDomain | analysis ⓘ |
| namedAfter | Leopold Kronecker ⓘ |
| relatedTo |
Tauberian theorems
ⓘ
surface form:
Abel’s theorem
Tauberian theorems ⓘ Tauberian theorems ⓘ
surface form:
Toeplitz theorem
|
| relates |
convergence of series
ⓘ
weighted averages of partial sums ⓘ |
| role |
auxiliary result in ergodic theory
ⓘ
technical lemma in harmonic analysis ⓘ |
| status |
classical result in summability theory
ⓘ
standard result in graduate real analysis ⓘ |
| topic |
Cesàro summation
ⓘ
surface form:
Cesàro-type averages
series of real numbers ⓘ summability of series ⓘ |
| usedAs | tool in proving strong laws in probability and ergodic theory ⓘ |
| usedFor | control of averages in convergence proofs ⓘ |
| usedIn |
Tauberian theorems
ⓘ
surface form:
Tauberian theory
proofs of ergodic theorems ⓘ study of Fourier series ⓘ summability methods ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Kronecker’s lemma Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.