Dear all,(adsbygoogle = window.adsbygoogle || []).push({});

I need a simple proof of the following:

Let [tex]u \in C(\mathbb{R}^3)[\tex] and [tex]\|u\|_{L^1(\mathbb{R}^3)} = 1[\tex]. For [tex]\lambda \geq 1[\tex], let us define the

transformation [tex]u\mapsto u_{\lambda}[\tex], where [tex] u_{\lambda}(x)={\lambda}^3 u(\lambda x)[\tex]. It is clear that

[tex]\|u_{\lambda}\|_{L^1(\mathbb{R}^3)} = \|u\|_{L^1(\mathbb{R}^3)} =1[\tex]. \\

How can I prove that

[tex]\lim_{\lambda\rightarrow\infty} u_{\lambda}(x)=\delta(x),[\tex] where [tex]\delta(x)[\tex] is the Dirac Delta function and

the limit is taken in the sense of distributions.

Thank you in advance.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Dirac Delta function as a limit

Loading...

Similar Threads - Dirac Delta function | Date |
---|---|

I Lebesgue Integral of Dirac Delta "function" | Nov 17, 2017 |

Dirac-delta function in spherical polar coordinates | Oct 7, 2017 |

I Understanding the Dirac Delta function | May 28, 2017 |

I Delta function in 2D | Jan 27, 2017 |

**Physics Forums - The Fusion of Science and Community**