Riemann–Lebesgue lemma

E47351

lemma mathematical theorem

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.

Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Riemann–Lebesgue lemma on locally compact abelian groups	1

Statements (46)

Predicate	Object
instanceOf	lemma ⓘ mathematical theorem ⓘ
appearsIn	textbooks on Lebesgue integration ⓘ textbooks on harmonic analysis ⓘ
appliesTo	Fourier analysis ⓘ surface form: Fourier series Fourier analysis ⓘ surface form: Fourier transform L¹ functions ⓘ integrable functions ⓘ
assumption	function belongs to L¹ with respect to the relevant measure ⓘ function is integrable in the Lebesgue sense ⓘ
conclusion	Fourier coefficients of an L¹ function vanish at infinity. ⓘ Fourier transform of an L¹ function vanishes at infinity. ⓘ
contrastWith	pointwise convergence of Fourier series ⓘ uniform convergence of Fourier series ⓘ
doesNotRequire	function to be continuous ⓘ function to be square-integrable ⓘ
domain	2π-periodic integrable functions ⓘ functions on the real line ⓘ functions on ℝⁿ ⓘ
field	Fourier analysis ⓘ harmonic analysis ⓘ real analysis ⓘ
generalization	Riemann–Lebesgue lemma self-linksurface differs ⓘ surface form: Riemann–Lebesgue lemma on locally compact abelian groups
holdsFor	absolutely integrable functions ⓘ compactly supported integrable functions ⓘ
implies	Fourier coefficients of an L¹ function form a null sequence. ⓘ Fourier transform of an L¹ function is a continuous function that tends to 0 at infinity. ⓘ
isWeakerThan	results that give rates of decay of Fourier coefficients ⓘ
namedAfter	Bernhard Riemann ⓘ Henri Lebesgue ⓘ
relatedTo	Fourier coefficient ⓘ Fourier inversion theorem ⓘ Fourier analysis ⓘ surface form: Fourier transform Lebesgue integration ⓘ surface form: Lebesgue integral L¹ space ⓘ Plancherel theorem for real reductive groups ⓘ surface form: Plancherel theorem trigonometric series ⓘ
statement	If f is in L¹(ℝⁿ), then its Fourier transform tends to 0 at infinity. ⓘ If f is integrable on [−π,π], then its Fourier coefficients tend to 0 as the frequency index tends to infinity. ⓘ
topicIn	graduate Fourier analysis courses ⓘ graduate real analysis courses ⓘ
typeOfLimit	asymptotic vanishing of frequency components ⓘ
usedIn	approximation theory ⓘ harmonic analysis on locally compact abelian groups ⓘ proofs of convergence results for Fourier series ⓘ signal processing theory ⓘ

Referenced by (4)

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

Riemann–Lebesgue lemma → generalization → Riemann–Lebesgue lemma self-linksurface differs ⓘ

this entity surface form: Riemann–Lebesgue lemma on locally compact abelian groups

Bernhard Riemann → knownFor → Riemann–Lebesgue lemma ⓘ

Friedrich → notableConcept → Riemann–Lebesgue lemma ⓘ

subject surface form: Friedrich Bernhard Riemann

Georg → notableWork → Riemann–Lebesgue lemma ⓘ

subject surface form: Georg Friedrich Bernhard Riemann