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### 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 ...
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### 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 $f(x)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...