Poisson summation formula
E300764
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Poisson summation formula canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T2815492 — 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: Poisson summation formula Context triple: [Euler–Maclaurin summation formula, relatedTo, Poisson summation formula]
-
A.
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.
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B.
Fourier inversion theorem
The Fourier inversion theorem is a fundamental result in harmonic analysis that guarantees, under suitable conditions, that a function can be exactly reconstructed from its Fourier transform.
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C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
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D.
Wiener–Khinchin theorem
The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
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E.
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.
- 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: Poisson summation formula Target entity description: The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
-
A.
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.
-
B.
Fourier inversion theorem
The Fourier inversion theorem is a fundamental result in harmonic analysis that guarantees, under suitable conditions, that a function can be exactly reconstructed from its Fourier transform.
-
C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
D.
Wiener–Khinchin theorem
The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
theorem in harmonic analysis ⓘ |
| appliesTo |
Schwartz functions on the real line
ⓘ
rapidly decreasing smooth functions ⓘ |
| category |
results in signal processing
ⓘ
theorems in analysis ⓘ theorems in number theory ⓘ |
| coreStatement | For suitable f, ∑_{n∈ℤ} f(n) = ∑_{k∈ℤ} ˆf(k) ⓘ |
| expresses | equality between spatial and frequency domain sums ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ mathematical physics ⓘ number theory ⓘ signal processing ⓘ |
| generalizedTo |
higher-dimensional Euclidean spaces
ⓘ
lattices in ℝ^n ⓘ locally compact abelian groups ⓘ |
| hasConsequence |
duality between time and frequency domains
ⓘ
periodicity relations between a function and its Fourier transform ⓘ |
| implies | Nyquist–Shannon sampling theorem under suitable hypotheses ⓘ |
| involves |
Dirac delta function
ⓘ
surface form:
Dirac comb
Fourier transform ⓘ periodization of functions ⓘ |
| namedAfter | Siméon Denis Poisson ⓘ |
| relatedTo |
Fourier series
ⓘ
Fourier transform on ℝ ⓘ Riemann–Siegel formula ⓘ theta transformation formula ⓘ |
| relates |
sum of a function over the integers
ⓘ
sum of the Fourier transform of a function over the integers ⓘ |
| requires | sufficient decay or regularity conditions on the function ⓘ |
| usedFor |
asymptotic analysis of sums
ⓘ
connecting discrete and continuous Fourier analysis ⓘ evaluating slowly convergent series ⓘ |
| usedIn |
Fourier series
ⓘ
surface form:
Fourier series expansions
aliasing analysis in signal processing ⓘ analysis of theta functions ⓘ crystallography and diffraction theory ⓘ derivation of the functional equation of the Riemann zeta function ⓘ heat kernel analysis ⓘ lattice point counting problems ⓘ modular forms ⓘ proofs in analytic number theory ⓘ quantum mechanics ⓘ sampling theory ⓘ solid-state physics ⓘ spectral 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: Poisson summation formula Description of subject: The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.