Riemann-Lebesgue lemma

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


The above result, commonly known as the Riemann-Lebesgue lemma, is of basic importance in harmonic analysis. It is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the assertion that the Fourier coefficients f^n of a periodic, integrable function f(x), tend to 0 as n±.

The proof can be organized into 3 steps.

Step 1. An elementary calculation shows that


for every interval I[a,b]. The propositionPlanetmathPlanetmath is therefore true for all step functionsPlanetmathPlanetmath with support (http://planetmath.org/SupportOfFunction) in [a,b].

Step 2. By the monotone convergence theoremMathworldPlanetmath, 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


where g and h are positive functions, integrable on [a,b].

Title Riemann-Lebesgue lemma
Canonical name RiemannLebesgueLemma
Date of creation 2013-03-22 13:08:04
Last modified on 2013-03-22 13:08:04
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Theorem
Classification msc 42A16