Riemann–Lebesgue lemma

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. It is of importance in harmonic analysis and asymptotic analysis.

If $f\in L^{1}[0,\infty )$ , then the Riemann–Lebesgue lemma also holds for the Laplace transform of $f$ , that is,

The Riemann–Lebesgue lemma holds in a variety of other situations.

Applications[edit]

The Riemann–Lebesgue lemma can be used to prove the validity of asymptotic approximations for integrals. Rigorous treatments of the method of steepest descent and the method of stationary phase, amongst others, are based on the Riemann–Lebesgue lemma.

, Chandrasekharan K. (1949). Fourier Transforms. Princeton University Press.

Bochner S.

"Riemann–Lebesgue Lemma". MathWorld.

Riemann–Lebesgue lemma

Applications[edit]

Bochner S.

Weisstein, Eric W.