Linked Questions

56 votes
9 answers

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 ...
jnm2's user avatar
  • 3,130
10 votes
2 answers

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 $f(x)$ ...
John's user avatar
  • 792
2 votes
2 answers

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\...
XaveryXavier's user avatar
2 votes
2 answers

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 ...
IamWill's user avatar
  • 3,945
1 vote
1 answer

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_{...
Student's user avatar
  • 388
4 votes
1 answer

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 ...
Daniels Krimans's user avatar
2 votes
1 answer

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 ...
Fork2's user avatar
  • 223
2 votes
1 answer

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 ...
Chaotic's user avatar
  • 163
0 votes
0 answers

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 ...
Abdul's user avatar
  • 147
1 vote
1 answer

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 ...
Daniels Krimans's user avatar
1 vote
1 answer

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 ...
Nothing's user avatar
  • 1,688