Riemann-Lebesgue lemma

.Let $f:[a,b]\to ℂ$ be a measurable function. If $f$ is$L^1$ integrable, that is to say if the Lebesgue integral of $|f|$ isfinite, then

$${\int }_a^{b}f(x)e^{inx}𝑑x\to 0,as\mathit{\hspace{1em}}n\to \pm \mathrm{\infty }.$$

The above result, commonly known as the Riemann-Lebesgue lemma, is ofbasic importance in harmonic analysis. It is equivalent to theassertion that the Fourier coefficients ${\widehat{f}}_n$ of a periodic, integrablefunction $f(x)$, tend to $0$ as $n\to \pm \mathrm{\infty }$.

The proof can be organized into 3 steps.

Step 1. An elementary calculation shows that

$${\int }_I{e}^{inx}𝑑x\to 0,as\mathit{\hspace{1em}}n\to \pm \mathrm{\infty }$$

for every interval $I\subset [a,b]$. The proposition is therefore truefor all step functions with support (http://planetmath.org/SupportOfFunction) in $[a,b]$.

Step 2.By the monotone convergence theorem, the proposition is true for allpositive 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$[a,b]$.