# Riemann-Lebesgue lemma

.
Let $f:[a,b]\to \u2102$ be a measurable function^{}. If $f$ is
${L}^{1}$ integrable, that is to say if the Lebesgue integral^{} of $|f|$ is
finite, then

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

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 ${\widehat{f}}_{n}$ of a periodic, integrable
function $f(x)$, tend to $0$ as $n\to \pm \mathrm{\infty}$.

The proof can be organized into 3 steps.

*Step 1.* An elementary calculation shows that

$${\int}_{I}{e}^{inx}\mathit{d}x\to 0,as\mathit{\hspace{1em}}n\to \pm \mathrm{\infty}$$ |

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 |
---|---|

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 |