# Tagged Questions

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### the continuous functions with norm

I'm having trouble trying to understand what does means the first expression in particular the last term in it should we add $\|f\|_{\infty} \leq \infty$ or what i can't see what is his role ...
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### How to build a compact support for a function

I was wondering if it is possible to build a distribution with compact support from a function. More precisely, consider a compact set $\mathbf{K}\subset\mathbb{R}^2\setminus\{0\}$, and a function ...
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### Derivatives of $|x|$

I wanted to calculate the first and second derivatives of the function $f(x)=|x|$, in order to verify that: $$f'(x)=\frac{|x|}{x}$$ and $$f''(x)=2\delta(x).$$ Can you help me?
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### Is this integral 0?

Let $\phi \in C_0^{\infty}(\Omega)$ with $\Omega = (0,1)\times(0,1)$. Let $u\in L_2(\Omega)$ defined by $u(x,y) = 1$ for $x>y, u(x,y) = 0$ for $x\leq y$ Is there a way we can conclude ...
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### Show $A$ is self-adjoint and $f= Au$ in weak sense.

Hoi, consider $L_2(\Omega)$ with $\Omega = (0,1)\times (0,1)$ and let $u\in L_2(\Omega)$ be defined as $u(x,y) = 1$ for $x>y$ and $u(x,y) = 0$ for $x\leq y$. Let $A = \partial_x^2 - \partial^2_y$ ...
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### Identifying 2 spaces of distributions

Hoi, I want to show that the space $C^{\infty}(\Omega)'$ of continuous linear functionals on $C^{\infty}(\Omega)$ can be identified with the subspace $\mathcal{E}'(\Omega)$ (distributions with ...
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### Inequalities for point distribution

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
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### distribution with point support

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
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### Fourier transform of locally integrable function

I have a question and I don't know if it is true. Is any locally integrable function a sum of an $L_{1}$ function and another nice function (perhaps, an $L_{2}$ function). This is related to the ...
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### Distributional limit

Let $u_t(x) = t^Ne^{itx}$ for $x\geqslant 0$ and $u_t(x)=0$ elsewhere. I want to calculate the distributional limit $\lim_{t\to\infty}u_t(x)$. How would one approach such problem. I just started a ...
I am trying to learn weak derivatives. In that we call $\mathbb{C}^{\infty}_{c}$ function as test function and we use this function in weak derivatives. I want to understand why these are called test ...
### Prove that $f$ does not have a weak derivative
Consider a function $f:\mathbb{R} \rightarrow [0,1 ]$ defined by: \$\begin{equation*} f(x)=\left\{ \begin{array}{rl}0 & \text{if } x\leq 0,\\ 1 & \text{if } x\geq 1, \\ 1/2 & \text{if } ...