# sinc is ${L}^{2}$

Our objective will be to prove the integral ${\int}_{\mathbb{R}}{f}^{2}(x)\mathit{d}x$ exists in the Lebesgue sense when $f(x)=\mathrm{sinc}(x)$.

The integrand is an even function and so we can restrict our proof to the set ${\mathbb{R}}^{+}$.

Since $f$ is a continuous function^{}, so will ${f}^{2}$ be and thus for every $a>0$, $f\in {L}^{2}([0,a])$.

Thus, if we prove $f\in {L}^{2}([\pi ,\mathrm{\infty}[)$, the result will be proved.

Consider the intervals ${I}_{k}=[k\pi ,(k+1)\pi ]$ and ${U}_{k}={\bigcup}_{i=1}^{k}{I}_{k}=[\pi ,(k+1)\pi ]$.

and the succession of functions ${f}_{n}(x)={f}^{2}(x){\chi}_{{U}_{n}}(x)$, where ${\chi}_{{U}_{n}}$ is the characteristic function^{} of the set ${U}_{n}$.

Each ${f}_{n}$ is a continuous function of compact support and will thus be integrable in ${\mathbb{R}}^{+}$. Furthermore ${f}_{n}(x)\nearrow {f}^{2}(x)$ (pointwise) in this set.

In each ${I}_{k}$,$0\le {f}^{2}(x)\le \frac{{\mathrm{sin}}^{2}(x)}{{(k\pi )}^{2}}$, for $k>0$.

So:

$\int}_{x\ge \pi}}{f}_{n}(x)\mathit{d}x={\displaystyle \sum _{k=1}^{n}}{\displaystyle {\int}_{k\pi}^{(k+1)\pi}}{\displaystyle \frac{\mathrm{sin}{(x)}^{2}}{{x}^{2}}}\mathit{d}x\le {\displaystyle \sum _{k=1}^{n}}{\displaystyle {\int}_{k\pi}^{(k+1)\pi}}{\displaystyle \frac{\mathrm{sin}{(x)}^{2}}{{(k\pi )}^{2}}}={\displaystyle \sum _{k=1}^{n}}{\displaystyle \frac{1}{2{k}^{2}\pi$
^{1}^{1}we have used the well known result ${\int}_{0}^{\pi}{\mathrm{sin}}^{2}(x)\mathit{d}x=\frac{\pi}{2}$

So: ${lim}_{n\to \mathrm{\infty}}{\int}_{x\ge \pi}{f}_{n}(x)\mathit{d}x\le {lim}_{n\to \mathrm{\infty}}{\sum}_{k=1}^{n}\frac{1}{2{k}^{2}\pi}$ and since the series on the right side converges^{2}^{2}asymptotic behaviour as ${k}^{-2}$ and ${f}_{n}\nearrow {f}^{2}$ we can use the monotone convergence theorem^{} to state that ${f}^{2}\in L([\pi ,\mathrm{\infty}[)$.

So we get the result that $\mathrm{sinc}\in {L}^{2}(\mathbb{R})$

Title | sinc is ${L}^{2}$ |
---|---|

Canonical name | SincIsL2 |

Date of creation | 2013-03-22 15:44:44 |

Last modified on | 2013-03-22 15:44:44 |

Owner | cvalente (11260) |

Last modified by | cvalente (11260) |

Numerical id | 9 |

Author | cvalente (11260) |

Entry type | Result |

Classification | msc 26A06 |