Dirac Delta function as a limit

In summary: The scalar product is a mathematical function that takes two vectors and produces a third vector. The test function is a function that takes one input and returns a scalar. The norm is a measure of how close two vectors are to each other. The scalar product of a distribution and the test function will produce a vector that is the limit of the distribution and the test function.
  • #1
gvenkov
3
0
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
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.
 
Last edited:
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  • #2
gvenkov said:
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 [itex]\lambda \geq 1[/itex], let us define the transformation [tex]u\rightarrow u_{\lambda}[/tex], where [itex] u_{\lambda}(x)={\lambda}^3 u(\lambda x)[/itex]. 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 [itex]\lim_{\lambda\rightarrow\infty} u_{\lambda}(x)=\delta(x),[/itex] where [itex]\delta(x)[/itex] 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] [ /tex] for equations (both without the spaces in the square brackets) instead of $ signs.
 
Last edited:
  • #3
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.
 
  • #4
Thak you very much for the help with the text.

George
 
  • #5
Dirac limit

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

1. What is the Dirac Delta function?

The Dirac Delta function, denoted by δ(x), is a mathematical function that is used to represent an infinitely narrow, infinitely tall spike at a specific point in the real number line. It is often referred to as the "unit impulse" or "unit spike" function.

2. How is the Dirac Delta function defined?

The Dirac Delta function is defined as a function that is equal to 0 for all values of x except at x=0, where it has a value of infinity (∞). Its integral over the entire real number line is equal to 1, making it a normalized function.

3. What is the significance of the Dirac Delta function as a limit?

The Dirac Delta function can be seen as a limit of a sequence of functions that become narrower and taller as the sequence goes to infinity. This limit can be used to represent a point mass or a point particle in physics and engineering, making it a useful tool in these fields.

4. How is the Dirac Delta function used in applications?

The Dirac Delta function is used in a wide range of applications, including signal processing, quantum mechanics, and probability theory. It is also commonly used in solving differential equations and as a tool for modeling physical systems.

5. What are some properties of the Dirac Delta function?

Some properties of the Dirac Delta function include:

  • It is an even function, meaning δ(x)=δ(-x).
  • It is a linear operator, meaning δ(ax+by)=aδ(x)+bδ(y).
  • It has the sifting property, meaning that when integrated over a function f(x), it picks out the value of f(0).
  • It has a scaling property, meaning that δ(ax)=1/|a|δ(x).
  • It has a shifting property, meaning that δ(x-x0)=0 when x≠x0 and δ(x-x0)=∞ when x=x0.

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