# 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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### Fundamental theorem of calculus for distributions

I want to show that if $f\in C([0,1])$ and the distributional derivative $f'$ on $(0,1)$ is in $L^1((0,1))$, then $$f(1) - f(0) = \int_0^1 f'(x)\,dx$$ I am having a lot of trouble getting started. ...
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### Can a schwartz class function be dominated by an exponential?

Given a function $f$ from Schwartz class, does there exist a constant $C$ such that $|f(x)|<Ce^{-|x|}$. For me its seems true, if it is not true, any counterexample would be very illustrative to me
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### how far the distribution from the uniform distribution

I have two discrete probability distributions $P$ and $Q$, where $P=(p_1,...,p_n)$ and $Q=(q_1,...,q_n)$, in addition I have uniform distribution $U=(\frac{1}{n},...,\frac{1}{n})$. The question is ...
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### $C_c^\infty(\Omega)\subseteq L^p(\Omega)$ for any open $\Omega$?

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$. Can we show that $$C_c^\infty(\Omega)\subseteq L^p(\Omega)\tag 1$$ for all $p\in [1,\infty]$? It's clear that $(1)$ holds if $\Omega$ has finite ...
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### Null space of the Laplacian operator?

(I guess the answer to my question is well-known in harmonic analysis, but I consider it in the framework of Schwartz distributions and in any dimension, and could not find a satisfactory answer.) ...
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### Proving that a function belongs in the space of tempered distributions

Let $a>0$ and define $$g(\xi):=\frac{\sin a\xi}{\xi(1+\xi^{2})}$$ I want to prove that $g\in\mathscr{S}'(\mathbb{R})$ and consquently that $g\in L^{1}(\mathbb{R})$ (but this implication is ...
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### Why is $f(x) \delta(x) = f(0)\delta(x)$ only true when $x=0$?

This is a follow up from a previous question asked by me. I know that $$\delta(x) = \begin{cases} 0 & \space \mathrm{for} \space x \ne 0 \\\infty&\ \mathrm{for} \space x = 0 \end{cases}$$ ...
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### Distribution theory book

I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the ...
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### I don't understand De Rham's theorem about the gradient of a distribution

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ and $$\mathfrak D(\Omega):=\left\{\Phi\in\mathcal D(\Omega)^d:\nabla\cdot\Phi=0\right\}$$ In a ...
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### Proof of the Helmholtz-Hodge decomposition

Let $\Omega\subseteq\mathbb R^3$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$G^2(\Omega):=\left\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega)^3\right\}$$...
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### Dirac functional embedding

I got the following statements to show. Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with ...
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### What is the gradient of a distribution?

Let $\Omega\subseteq\mathbb R^d$ be open and $\mathcal D(\Omega)$ be the set of $C^\infty(\Omega)$-functions with compact support equipped with a locally convex topology. Let $\mathcal D(\Omega)'$ ...
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### How to define a variable which is an integral involving cauchy principal value inside?

How to define a variable which is an integral involving cauchy principal value inside in any computer programming language? I want to know how to break down the procedure step by step from a ...
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### A doubt about the vectorial topology on $\mathcal{D}(\Omega)$

We denote with $\mathcal{U}_0$ the family of all subsets $U \in \mathcal{D}(\Omega)$ convex and balanced such that $U \cap \mathcal{D}_K(\Omega) \in \mathcal{T}_K$, where $\mathcal{T}_K$ is the ...
### How to show for a distribution $T$ and a test function $\varphi,~~T'[\varphi]\equiv -T[\varphi']\;?$
For a generalized function $T,$ we define $$T'[\varphi] ~≡~ −T[φ']~~~~~~\forall φ ∈ \mathcal D(Ω).$$ where $\mathcal D(\Omega)$ denotes the test function space. I'm not getting how they ...