The result is named for mathematicians Riemann and Henri Lebesgue, and is important in our understanding of Fourier Series and the Fourier Transform.
The Riemann-Lebesgue Lemma
If is integrable, which is to say that
then as .
In English, this means that the Fourier transform of an absolutely integrable function vanishes at infinity. This result
is important for resolving the issues of convergence of Fourier series and the asymptotic behavior
of Fourier transforms and Fourier-like integrals, such as those encountered in seismic imaging.
Proof , Case 1 - infinite limits of integration  =
Consider the forward Fourier transform
and make use of the fact that
Rewriting this expression slightly,
and changing variables by replacing by , we obtain
We may write, combining this result with the initial representation of the Fourier transform, as
Applying the Cauchy-Schwartz inequality, we may write
Thus we may write, in terms of the Bachman-Landau order symbol that as
Rate of Convergence
Of course, this does not tell us anything about the rate at which tends to zero. We only know that the Fourier transform of
an -integrable function vanishes as
If vanishes smoothly, which is to say that is a continuous function that is -times differentiable
and that and all of its derivatives vanish as
Then we may perform repetitive integration by parts -times to yield
The function which is the -th derivative of is also -integrable, meaning that we can estimate the
integral as being by the Riemann-Lebesgue Lemma, and write that
If is then decays faster than any power of
- ↑ Lighthill, M. J. (1962). Fourier analysis and generalised functions. Cambridge University Press, p.46.