In mathematics, the Riemann-Lebesgue lemma is of importance in harmonic analysis and asymptotic analysis. It is named after Bernhard Riemann and Henri Lebesgue.

Intuitively, the lemma says that if a function oscillates rapidly around zero, then the integral of this function will be small. The integral will approach zero as the number of oscillations increases.

Definition

Let f:[a,b] → C be a measurable function. If f is L¹ integrable, that is to say if the Lebesgue integral of |f| is finite, then

as .

This is equivalent to the assertion that the Fourier coefficients

of a periodic, integrable function f(x), tend to 0 as n → ± ∞.

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

The proof can be organized into 3 steps.

Step 1. An elementary calculation shows that

as

for every interval I ⊂ [a, b]. The proposition is therefore true for all step functions with support in [a, b].

Step 2. By the monotone convergence theorem, the proposition is true for all positive functions, integrable on [a, b].

Step 3. Let f be an arbitrary measurable function, integrable on [a, b]. The proposition is true for such a general f, because one can always write f = g − h where g and h are positive functions, integrable on [a, b].