generalized Riemann-Lebesgue lemma

Generalized Riemann-Lebesgue lemmaFernando Sanz Gamiz

Lemma 1.

Let h:RC be a bounded measurable functionMathworldPlanetmath. If h satisfies the averaging condition




with -<a<b<+ for any fL1[a,b]


Obviously we only need to prove the lemma when both h and f are real and 0=a<b<.

Let 𝟏[a,b] be the indicator functionPlanetmathPlanetmath of the interval [a,b]. Then


by the hypothesisMathworldPlanetmath. Hence, the lemma is valid for indicators, therefore for step functionsPlanetmathPlanetmath.

Now let C be a bound for h and choose ϵ >0. As step functions are dense in L1, we can find, for any fL1[a,b], a step function g such that f-g1<ϵ, therefore

limω|abf(t)h(ωt)𝑑t| limωab|f(t)-g(t)||h(ωt)|𝑑t+limω|abg(t)h(ωt)𝑑t|

because limω|abg(t)h(ωt)𝑑t|=0 by what we have proved for step functions. Since ϵ is arbitrary, we are done.

Title generalized Riemann-Lebesgue lemma
Canonical name GeneralizedRiemannLebesgueLemma
Date of creation 2013-03-22 17:06:03
Last modified on 2013-03-22 17:06:03
Owner fernsanz (8869)
Last modified by fernsanz (8869)
Numerical id 13
Author fernsanz (8869)
Entry type Theorem
Classification msc 42A16
Related topic RiemannLebesgueLemma
Related topic FourierCoefficients
Related topic Integral2