How to Prove the Dirac Delta Function as a Limit?

[gvenkov]

Dear all,

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<br /> transformation [tex]u\mapsto u_{\lambda}[\tex], where [tex]u_{\lambda}(x)={\lambda}^3 u(\lambda x)[\tex]. It is clear that<br /> [tex]\|u_{\lambda}\|_{L^1(\mathbb{R}^3)} = \|u\|_{L^1(\mathbb{R}^3)} =1[\tex]. \\<br /> How can I prove that<br /> [tex]\lim_{\lambda\rightarrow\infty} u_{\lambda}(x)=\delta(x),[\tex] where $$\delta(x)[\tex] is the Dirac Delta function and<br /> the limit is taken in the sense of distributions.<br /> <br /> Thank you in advance.$$[/tex][/tex][/tex][/tex][/tex][/tex][/tex]

 

[cristo]

Dear all,

I need a simple proof of the following:

Let $$u \in C(\mathbb{R}^3)$$ and $$\|u\|_{L^1(\mathbb{R}^3)} = 1$$. For $\lambda \geq 1$, let us define the transformation $$u\rightarrow u_{\lambda}$$, where $u_{\lambda}(x)={\lambda}^3 u(\lambda x)$. It is clear that $$\|u_{\lambda}\|_{L^1(\mathbb{R}^3)} = \|u\|_{L^1(\mathbb{R}^3)} =1$$. How can I prove that $\lim_{\lambda\rightarrow\infty} u_{\lambda}(x)=\delta(x),$ where $\delta(x)$ is the Dirac Delta function and the limit is taken in the sense of distributions.

Thank you in advance.

I've just tidied up your LaTex before I try and read it. Note that on the forum to get tex to show either use [itex][ /itex] tags for inline tex or $$[ /tex] for equations (both without the spaces in the square brackets) instead of signs.$$[/itex]

 

[HallsofIvy]

Also, you have to use /tex, not \tex, to end LaTex.


To prove that the limit is the delta function, look at the limit of the integral of each of your functions times some test function f(x) dx.

 

[gvenkov]

Thak you very much for the help with the text.

George

 

[gvenkov]

Dirac limit

To do this I have to define the scalar product of a distribution and an arbitrary test function, and an appropriate norm.