# Tagged Questions

Use this tag for questions about distributions (or generalized functions). For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

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### Second distributional derivative of cosine

I need to compute second distributional derivative of the function $$g(x) = cos|x-2|,$$ but I'm not sure about my solution. \begin{align} \left<g'', \varphi \right> = \left<g, \varphi''\...
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### Convolution of Schwartz and test function approximated by partition of unity.

Let $\rho\in\mathscr{D}$, $0\leq\rho\leq 1, \rho(0) = 1$, and $\sum_{n\in\mathbb{Z}^d}$ $\rho(x-n) = 1$. Denote, $\rho_{n,\epsilon}() = \tau_n\rho(\frac{x}{\epsilon})$, where $\tau$ is the translation ...
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### $r:\mathscr{S'}\rightarrow\mathscr{D'}$, $u\mapsto u|_{\mathscr{D}}$ is not a topological embedding

Show that $r:\mathscr{S'}\rightarrow\mathscr{D'}$, $u\mapsto u|_{\mathscr{D}}$ is not a topological embedding. For this problem, would it suffice to construct a sequence $\{u_n\}$ in $\mathscr{D'}$ ...
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### If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$.

If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$. I am not quite sure how to start this problem. ...
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### Prove that $e^x$ is not a tempered distribution on $\mathbb{R}$

Consider the following sequence of functions $\psi_n(x) = e^{-(1+\varepsilon)x} \dfrac{1_{|x|\leq n}}{n}$. Clearly, $|\psi_n^{(m)}(x)|\leq\dfrac{(1+\varepsilon)^m}{n}$. Hence, the $\psi_n$-s are ...
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### How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$?

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$? Where $\mathcal{D}(\Omega)$ is the space of test functions with support compact and $\mathcal{D}'(\Omega)$ is the ...
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### Distributional derivatives

I need to compute derivatives as distributions of following functions: $f(x) =$ $|x|$ $|x^2 - 1|$ $\mathrm{sgn}(x)$ $4$ Where $f : \mathbb{R} \to \mathbb{R}$. ad 1) $|x|$ is continuous, so it ...
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### Convolution of Schwartz Function and Distribution of Compact Support

From Stein-Shakarchi Functional Analysis Chapter 3 Exercise 12 and Exercise 13. I'm having trouble proving that: If $F_1$ is a distribution with compact support and $\varphi\in \mathcal{S}$ is a ...
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### Convolution of two distributions

Consider the convolution product: $$H(x)\ast\operatorname{Pf}\dfrac{H(x)}{x},$$ where $\operatorname{Pf}$ denotes pseudo function. This means, that $\operatorname{Pf}\dfrac{H(x)}{x}$ is, as defined ...
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### Do tempered distributions form a topological subspace of the space of distributions?

I'm learning about distributions and tempered distributions. From what I understand, by "enlarging" the space of test functions $\mathcal{D}$ to the Schwarz space $\mathcal{S}$ and correspondingly "...
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### $\Lambda\psi$=0 if $(D^{\alpha}\psi)(x)$ for every $x\in$ supp $\Lambda$ and every multi index $\alpha$

I can show that $\Lambda\psi$=0 if $(D^{\alpha}\psi)(x)$ for every $x\in$ supp $\Lambda$ and every multi index $\alpha$ given the support of $\Lambda$ is compact. But how one extend this argument for ...
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### Continuous linear functional on $C^{\infty}_{loc}(\mathbb{R}^d)$ is a distribution.

Prove that every continuous linear functional on $C^{\infty}_{loc}(\mathbb{R}^d)$ is of the form $\Lambda\mapsto\Lambda f$ for some distribution $\Lambda$ with compact support. I am stuck at this ...
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### Positive distribution $\Lambda$ as positive Radon measure

Exercise 4 of Chapter 6 in Rudin's Functional Analysis states that every "positive" distribution $\Lambda\in D^{'}(\Omega)$, i.e, $\Lambda\psi\geq 0$ whenever $\psi\in D(\Omega)$, is a positive ...
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There exists a standard method to build test functions? I would like to construct a sequence of functions $\phi_{m} \in C^{\infty}_{c}(B_{1})$ such that $\phi_{m} \to \psi{u}^{-2}$, where $\psi \in H^... 1answer 30 views ### Is there is notion of Fourier transform of distribution? We note that every tempered distribution is a distribution. Can we find a example of distribution which is NOT a tempered distribution? Can we talk of Fourier transform of that distribution?... 0answers 26 views ### Convergence in convolution This is an exercise in Rudin's Functional analysis, which is 6.23. The problem:$f_i \in L_{loc}^1(\mathbb{R}^n), \ \lim\limits_{i \to \infty}(f_i*\phi)(x)$exists,$\forall \phi \in \mathscr{D}, \ x ...
I'm trying to solve the following exercise regarding limits of distributions: Establish the following limit (on the distributional sense) \lim_{t\to 0\pm}\ln (\tau + it) = \ln |\tau| + i\pi H(-...