Riemann-Lebesque lemma

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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 [1] =

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

as

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

References

  1. Lighthill, M. J. (1962). Fourier analysis and generalised functions. Cambridge University Press, p.46.