Linked Questions
11 questions linked to/from Dirac's $\delta$ distribution smooth approximation
56
votes
9
answers
23k
views
Are there other kinds of bump functions than $e^\frac1{x^2-1}$?
I've only seen the bump function $e^\frac1{x^2-1}$ so far. Where could I find examples of functions $C^∞$ on $\mathbb{R}$ that are zero everywhere except on $(-1,1)$?
Are there others that do not ...
- 3,130
10
votes
2
answers
10k
views
How to smoothly approximate a sign function
I have a function that defined as following
$$f(x) =
\begin{cases}
1, & \text{if $x > 0$ } \\
0, & \text{if $x=0$ } \\
-1, & \text{if $x<0$ }
\end{cases}$$
In practice, the $f(x)$ ...
- 792
2
votes
2
answers
1k
views
Dirac delta as a limit of sequence of functions
I have problem to proof that dirac delta function can be represented as following limits:
$$ \lim_{\varepsilon\rightarrow 0}\frac{\varepsilon}{\pi(x^2+\varepsilon^2)}=\delta(x) $$
$$ \lim_{\varepsilon\...
- 45
2
votes
2
answers
885
views
Convergence to Delta Dirac Distribution
This question derived from my previous question. When I took a course on the theory of distributions, I was first introduced to the Dirac delta as an usual distribution, that is, as a linear ...
- 3,945
1
vote
1
answer
1k
views
Dirac Delta limiting representation
This question is about making consistent the definition and properties of the Dirac Delta function from two different sources.
In this online textbook, the following definition is given.
$$\delta_{...
- 388
4
votes
1
answer
219
views
On computing $\nabla^2 (1/4 \pi |x|)$ in three dimensions using mollifiers.
I am going through some notes about how to compute that $\nabla^2 G = \nabla^2 (1/4 \pi |x|) = - \delta(x)$ in a sense of distributions, where $x \in \mathbb{R}^3$. Notes are not expected to be ...
- 1,453
2
votes
1
answer
440
views
Proof on delta sequences
In a coursetext of mine there is a theorem that says: when $f(x)$ is locally integrable ($f(x)\in\text{L}^1_{\text{loc}}(\mathbb{R}^d$), $f(x)\geq0 \,\forall x$ and $\int f(x) dx=1$ then it follows ...
- 223
2
votes
1
answer
415
views
Simplest smooth ($C^{\infty}$) approximation to Dirac's $\delta$ with bounded support.
I'm looking for a function $f(x)$ that approximates Dirac's $\delta$ distribution that has the following properties:
$f\in C^\infty(\mathbf R)$, it has finite derivatives of all orders.
$f$ has a ...
- 163
0
votes
0
answers
199
views
Poisson Kernel in the upper half-plane is an identity approximation
I read that the Poisson Kernel for the upper half-plane, $K_y(x)=\frac{1}{\pi}\frac{y}{y^2+x^2}$ is an approximation to the identity. The text states this without proof and I am hoping to see a proof ...
- 147
1
vote
1
answer
147
views
On computing distribution $\partial_i \partial_j (1/|x|)$ in three dimensions using regularization $1/\sqrt{|x|^2 + \varepsilon^2}$.
Motivated by questions and answers in here and here, I would like to understand how to interpret and prove the following identity in a sense of distributions, where $x \in \mathbb{R}^3$ (assuming it ...
- 1,453
1
vote
1
answer
136
views
Finding convergence to dirac delta function in two dimension.
I have a function $\nabla^2\ln(x^2+y^2+\epsilon) = \left(\frac{4\epsilon}{x^2+y^2+\epsilon}\right)^2$, where it converges to zero if $(x,y)\neq 0 $ and diverges if $(x,y) = 0$ as $\epsilon\rightarrow ...
- 1,688