Tauberian theorems

E451517

class of mathematical theorems

Tauberian theorems are results in mathematical analysis that connect the behavior of transformed series or integrals (such as those summed by Abel or Cesàro methods) back to the asymptotic behavior or convergence of the original sequences or series.

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All labels observed (7)

Label	Occurrences
Tauberian theorems canonical	5
Abel’s theorem	1
Hardy–Littlewood Tauberian theorem	1
Perron’s formula	1
Tauberian theory	1
Toeplitz theorem	1
Wiener Tauberian theorem	1

Statements (46)

Predicate	Object
instanceOf	class of mathematical theorems ⓘ
appliesTo	Dirichlet series NERFINISHED ⓘ Fourier series NERFINISHED ⓘ integral transforms ⓘ power series ⓘ
characterizes	when summability implies ordinary convergence ⓘ
concerns	boundary behavior of transforms ⓘ summability of sequences ⓘ summability of series ⓘ
contrastsWith	Abelian theorems NERFINISHED ⓘ
field	asymptotic analysis ⓘ mathematical analysis ⓘ summability theory ⓘ
generalizes	results of Alfred Tauber ⓘ
goal	recover original behavior from transformed behavior ⓘ
hasSubtype	Abelian–Tauberian theorems ⓘ Delange Tauberian theorems NERFINISHED ⓘ Hardy–Littlewood Tauberian theorems NERFINISHED ⓘ Ikehara Tauberian theorem NERFINISHED ⓘ Ingham Tauberian theorems NERFINISHED ⓘ Karamata Tauberian theorems NERFINISHED ⓘ Wiener Tauberian theorems NERFINISHED ⓘ
historicalOrigin	early 20th century ⓘ
namedAfter	Alfred Tauber NERFINISHED ⓘ
relatedConcept	Abelian theorems NERFINISHED ⓘ Hardy–Littlewood–Karamata theory of regular variation NERFINISHED ⓘ Wiener’s Tauberian theorem NERFINISHED ⓘ regular variation ⓘ
relates	asymptotic behavior of sequences ⓘ asymptotic behavior of series ⓘ convergence of sequences ⓘ convergence of series ⓘ summability methods ⓘ
typicalCondition	growth restrictions on coefficients ⓘ regularity conditions on transforms ⓘ
usedIn	analytic number theory ⓘ complex analysis ⓘ harmonic analysis ⓘ operator theory ⓘ prime number theory ⓘ probability theory ⓘ renewal theory ⓘ
uses	Abel summation method ⓘ Cesàro summation method NERFINISHED ⓘ Fourier transform ⓘ Laplace transform NERFINISHED ⓘ

Referenced by (11)

Full triples — surface form annotated when it differs from this entity's canonical label.

Divergent Series → topic → Tauberian theorems ⓘ

Kronecker’s lemma → usedIn → Tauberian theorems ⓘ

this entity surface form: Tauberian theory

Kronecker’s lemma → relatedTo → Tauberian theorems ⓘ

this entity surface form: Toeplitz theorem

Kronecker’s lemma → relatedTo → Tauberian theorems ⓘ

this entity surface form: Abel’s theorem

Kronecker’s lemma → relatedTo → Tauberian theorems ⓘ

John Edensor Littlewood → knownFor → Tauberian theorems ⓘ

this entity surface form: Hardy–Littlewood Tauberian theorem

The Fourier Integral and Certain of Its Applications → subject → Tauberian theorems ⓘ

The Fourier Integral and Certain of Its Applications → relatedConcept → Tauberian theorems ⓘ

this entity surface form: Wiener Tauberian theorem

Multiplicative Number Theory → methodUsed → Tauberian theorems ⓘ

this entity surface form: Perron’s formula

Dirichlet series → isToolFor → Tauberian theorems ⓘ