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In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, is of importance in harmonic analysis and asymptotic analysis.
Contents
The lemma says that the Fourier transform or Laplace transform of an L1 function vanishes at infinity.
Statement
If ƒ is L1 integrable on Rd, that is to say, if the Lebesgue integral of |ƒ| is finite, then the Fourier transform of ƒ satisfies
Other versions
The Riemann–Lebesgue lemma holds in a variety of other situations.
Applications
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.
Proof
We'll focus on the one-dimensional case, the proof in higher dimensions is similar. Suppose first that ƒ is a compactly supported smooth function. Then integration yields
If ƒ is an arbitrary integrable function, it may be approximated in the L1 norm by a compactly supported smooth function g. Pick such a g so that ||ƒ − g||L1 < ε. Then
and since this holds for any ε > 0, the theorem follows.