Riemann-Lebesgue lemma

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The Riemann-Lebesgue lemma states that the integral of a function like the above is small.

In mathematics, the Riemann-Lebesgue lemma, also called Mercer's theorem, 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.

1 Definition
2 Applications
3 Proof
4 References

[edit] 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

$\int^b_a f(x) e^{inx}\,dx \rightarrow 0$ as $\quad n\rightarrow \pm\infty$ .

This is equivalent to the assertion that the Fourier coefficients

$\hat{f}_n$

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

[edit] 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.

[edit] Proof

The proof can be organized into 3 steps.

Step 1. An elementary calculation shows that

$\int_I e^{inx}\,dx \rightarrow 0$ as $\quad n\rightarrow \pm\infty$

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].