Weierstrass approximation theorem
E110605
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Weierstrass approximation theorem canonical | 5 |
| Stone–Weierstrass theorem | 1 |
| approximation theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T940258 — 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: Weierstrass approximation theorem Context triple: [Karl Weierstrass, notableFor, Weierstrass approximation theorem]
-
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.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
C.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
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D.
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).
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E.
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.
- 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: Weierstrass approximation theorem Target entity description: The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
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.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
C.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
-
D.
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).
-
E.
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.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
approximation theorem
ⓘ
theorem in real analysis ⓘ |
| alternativeProofMethod |
Bernstein polynomials
ⓘ
convolution with approximate identities ⓘ |
| appliesTo | real-valued continuous functions on compact subsets of R (via reduction to intervals) ⓘ |
| approximationFamily | polynomials with real coefficients ⓘ |
| approximationType | uniform approximation ⓘ |
| assumption | interval is compact in R ⓘ |
| category |
density result
ⓘ
existence theorem ⓘ |
| codomainCondition | real-valued function ⓘ |
| conclusion | there exists a polynomial whose uniform distance to the function is less than epsilon ⓘ |
| consequence | continuous functions on [a,b] can be approximated arbitrarily well by elementary functions (polynomials) ⓘ |
| doesNotRequire |
differentiability of the function
ⓘ
periodicity of the function ⓘ |
| domainCondition |
domain is a closed interval [a,b] in R
ⓘ
function is continuous ⓘ |
| field |
approximation theory
ⓘ
real analysis ⓘ |
| generalizationOf | Stone–Weierstrass theorem ⓘ |
| historicalYear | 1885 ⓘ |
| implies | polynomials are dense in C([a,b]) with respect to the uniform norm ⓘ |
| inspired | development of abstract approximation theory ⓘ |
| isFundamentalResultIn |
approximation theory curriculum
ⓘ
undergraduate real analysis ⓘ |
| namedAfter | Karl Weierstrass ⓘ |
| norm | supremum norm ⓘ |
| originalProofMethod | trigonometric polynomials and substitution ⓘ |
| quantifier |
for every continuous function on [a,b]
ⓘ
for every epsilon greater than 0 ⓘ |
| relatedTo |
Bernstein polynomials
ⓘ
Fourier series ⓘ
linked to:
Fourier analysis
Runge approximation theorem ⓘ Stone–Weierstrass theorem ⓘ |
| space | C([a,b]) ⓘ |
| statement | Every continuous real-valued function on a closed and bounded interval can be uniformly approximated by polynomials. ⓘ |
| topologicalStatement | polynomials are dense in the space of continuous functions on a compact interval ⓘ |
| usedIn |
construction of polynomial interpolants
ⓘ
functional analysis ⓘ numerical analysis ⓘ theory of function approximation ⓘ |
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: Weierstrass approximation theorem Description of subject: The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Weierstrass approximation theorem
linked to: Weierstrass approximation theorem