# Riemann-Lebesgue lemma

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

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

The above result, commonly known as the Riemann-Lebesgue lemma, is of basic importance in harmonic analysis. It is equivalent to the assertion that the Fourier coefficients $\hat{f}_{n}$ of a periodic, integrable function $f(x)$, tend to $0$ as $n\rightarrow\pm\infty$.

The proof can be organized into 3 steps.

Step 1. An elementary calculation shows that

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

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

Title Riemann-Lebesgue lemma RiemannLebesgueLemma 2013-03-22 13:08:04 2013-03-22 13:08:04 rmilson (146) rmilson (146) 5 rmilson (146) Theorem msc 42A16