The Riemann-Lebesgue Lemma states that the Fourier transform, or the Fourier coefficients, of an integrable function must tend to zero as the frequency parameter grows without bound. Named for Bernhard Riemann and Henri Lebesgue, who each proved versions of the result in different settings, it is a basic tool of harmonic analysis guaranteeing that highly oscillatory integrals against an integrable function vanish in the limit.
Facts
StatementThe Riemann-Lebesgue lemma states that the Fourier transform, or the Fourier coefficients, of an integrable (L1) function tends to zero as the frequency parameter grows without bound, so the transform of an integrable function vanishes at infinity. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Sources
1. Riemann-Lebesgue Lemma (Wikipedia)
Wikimedia Foundationlead paragraph, first sentenceQuote, lead paragraph, first sentence
In mathematics, the Riemann-Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity.
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