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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2
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1answer
29 views

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. ...
1
vote
2answers
44 views

What is the divergence of a distribution?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ If $p\in \mathcal D'(\Omega)$, then $$\frac{\partial p}{\partial x_i}(\phi):=-p\left(\frac{\...
0
votes
1answer
46 views

If $p$ is a distribution, what is the meaning of the claim $\nabla p\in L^p(\Omega)^d$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 1$ I've seen the following Lemma (without a proof) in a paper and don't understand how I ...
0
votes
2answers
50 views

If $p,q$ are distributions with $\partial_ip=\partial_iq$, then $p=q$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$\frac{\partial p}{\partial x_i}(\phi):=-p\left(\frac{\partial \phi}{\partial x_i}\right)\;\;\;\...
0
votes
1answer
56 views

Has the distributional Laplacian $\Delta f:C_c^\infty(\Omega)'\to C_c^\infty(\Omega)'$ a unique extension in $H_0^1(\Omega)'$?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega)$ and $$H=\overline{\mathcal D}^{\langle\;\cdot\;,\;\cdot\;\rangle_H}\tag 1$$ with $$\langle\phi,\psi\rangle_H:...
0
votes
0answers
31 views

Computing the mean and variance of the ratio of two Laplace variables

I know that Laplacian distribution function is defined as follow $$ f(x)=\frac{b}{2}\exp(-b|x-\mu|) $$ Also, I know that the mean and variance for the ratio between two normal variables like $$c=\...
1
vote
1answer
25 views

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
1
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0answers
36 views

Relationship between the distributional Laplacian and the weak Laplacian

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the $L^2(\Omega)$- or $L^2(\Omega,\mathbb R^d)$-inner product (depending on the context) $\mathcal ...
2
votes
1answer
28 views

Variation of parameters for ODES with distributions as coefficients

In my work, I encounter the following type of equation. Consider a non-homogeneous system \begin{equation} X'=A(t)X+f(t),\;\;\;\;\;\;\;\;(1) \end{equation} where $X$ is a $n$-dimensional vector valued ...
1
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0answers
26 views

Product of currents

De Rham currents are for differential forms as distributions are to (smooth) functions. There is a notion of exterior product for differential forms: I wonder whether there is an algebra structure on ...
2
votes
1answer
28 views

Composition with Second Derivative of Dirac Distribution

Here's a nasty question that came up on an old qualifying exam that I'm helping students study. Let $T = \delta^{\prime\prime}(\cos(x))$ and let $\varphi = e^{-x^2}$. Evaluate $\langle T,\...
2
votes
2answers
30 views

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 ...
1
vote
4answers
52 views

$C_c^\infty(\Omega)\subseteq L^p(\Omega)$ 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 ...
1
vote
1answer
41 views

Distributional limit of $f_n = e^{x/n}$

Find the distributional limit of $f_n = e^{x/n}$. That means that I have to find the limit of $T_{f_n}$. I proceeded as follows. Let $\phi \in \mathcal D$ be a test function. Then: $$\langle T_{f_n},...
1
vote
0answers
29 views

Deducing an equality involving Fourier transform of Schwarz functions

For $\psi,\phi\in\mathscr{S}(\mathbb{R}^n)$ we have $$\psi(0)\int_{\mathbb{R}^n}\mathscr{F}\phi(\xi) \, d\xi = \phi(0) \int_{\mathbb{R}^n} \mathscr{F}\psi(\xi) \, d\xi$$ I want to prove that this ...
0
votes
1answer
37 views

Volterra Operator on Sobolev Space

I stumpled over the following result in a script: Let $1 \leq p < \infty$ and $f \in L_p[a,b]$. Define the Volterra operator as $$Vf(t) = \int_a^t f(s) ds.$$ Then we have $Vf \in W^{1, p}[a,b]$ ...
1
vote
1answer
49 views

Definition of the Laplacian as an operator from $H_0^1(\Omega)$ to $H_0^1(\Omega)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $f\in L^2(\Omega)$ and $$\langle f\rangle:=\left.\langle\;\cdot\;,f\rangle_{L^2(\Omega)}\right|_{\...
1
vote
1answer
43 views

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.) ...
0
votes
1answer
42 views

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 ...
4
votes
3answers
209 views

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 $$\delta(x) = \begin{cases} 0 & \space \mathrm{for} \space x \ne 0 \\\infty&\ \mathrm{for} \space x = 0 \end{cases} $$ ...
9
votes
6answers
2k views

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 ...
4
votes
2answers
260 views

Definition of the convolution with tempered distributions and Schwartz function

In the book where I'm studying there is the following exercise. If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=...
0
votes
1answer
28 views

Can we talk about the adjoint of a linear operator defined on a distribution space?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ and $$...
0
votes
0answers
20 views

Fourier transforms of some homogeneous functions

In $2D$, what are the Fourier transforms (in the sense of distributions) of functions of the form $x_i/|x|^2, 1/|x|, x_i/|x|, x_i x_j /|x|^2$, where $i = 1,2$. They are homogeneous and integrable ...
2
votes
1answer
47 views

How to show that a Schwartz distribution is in a Lebesgue or Sobolev space?

It is known that all $L^p$ spaces (and, consequently, all $W^{s,p}$ spaces) can be embedded in the space of Schwartz distributions $\mathcal D '$. There is a problem, though: how do I check whether ...
1
vote
0answers
30 views

Prove that $\left.F\right|_{\left\{ϕ∈C_c^∞(Ω,ℝ^d):∇⋅ϕ=0\right\}}=0⇔∃p∈C_c^∞(Ω)$ with $F=∇p$, for all $F∈H_0^1(Ω,ℝ^d)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$, $$H:=\...
3
votes
0answers
101 views

If $F∈H^{-1}(Ω,ℝ^d)$ and $∃p∈\mathcal D'(Ω):\left.F\right|_{\mathcal D(Ω,ℝ^d)}=∇p$, then $∃\overline p∈H^{-1}(Ω):F=∇\overline p$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ and $\mathcal D(\Omega,\mathbb R^d):=C_c^\infty(\Omega,\mathbb R^d)$ $H^{-1}(\Omega):=H_0^1(\Omega)'$...
3
votes
2answers
63 views

Relationship between $C_c^\infty(\Omega,\mathbb R^d)'$ and $H_0^1(\Omega,\mathbb R^d)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$...
1
vote
1answer
40 views

Property singular support of the convolution of distributions

Let $u \in \mathcal{D}'(\Omega)$ and $U \subset \Omega$ open. By definition we say that $u \in \mathcal{E}(U)$ if $\exists u(x) \in \mathcal{E}(U)$ such that \begin{align*} \displaystyle \langle \...
1
vote
1answer
59 views

Is the restriction $f$ of $F\in H^{-1}(\Omega,\mathbb R^d)$ to $C_c^\infty(\Omega,\mathbb R^d)$ a distribution?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$...
2
votes
0answers
27 views

If $p$ is a distribution such that $\nabla p$ is regular, then $p$ must be regular too

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 1$ Each $f\in L^1_{\text{loc}}(\Omega)$ can be identified with $\langle f\rangle\in\mathcal ...
1
vote
0answers
36 views

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 ...
0
votes
0answers
22 views

Characterization of the Gradient of a Distribution

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ (without topology) $u:\mathcal D(\Omega)\to\mathbb R$ is called distribution on $\Omega$ $:\...
0
votes
0answers
15 views

Characterization of a set occurring in the Helmholtz-Hodge decomposition

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 2$ Each $f\in L^1_{\text{loc}}(\Omega)$ can be identified with $\langle f\rangle\in\mathcal ...
4
votes
0answers
56 views

Proof of the Helmholtz-Hodge decomposition

Let $\Omega\subseteq\mathbb R^3$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$G^2(\Omega):=\left\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega)^3\right\}$$...
0
votes
1answer
33 views

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 ...
0
votes
1answer
43 views

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 $\mathcal D(\Omega)'$ ...
0
votes
0answers
21 views

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 ...
2
votes
2answers
69 views

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 ...
3
votes
1answer
97 views

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 ...
0
votes
2answers
24 views

Representation of functional on overlapping areas

I have given a functional $l$ on $C_c^\infty(\mathbb{R}^n)$. Now let's assume that for any $p \in \mathbb{R}^n$ we have a neighborhood $V_p$ and a $2\pi$-periodic $C^\infty$-function $u_p$ on $\mathbb{...
1
vote
1answer
36 views

Derivative of principal value distribution $1/x^2$ is equal to finite part distribution $-1/x^2$?

Finite part (Partie finie) of the mapping $x \mapsto \frac{1}{{{x^2}}}$ is a regular distribution defined by $$\left\langle {{\text{Pf}}\frac{1}{{{x^2}}},\varphi } \right\rangle = \mathop {\lim }\...
1
vote
1answer
24 views

Distributional derivatives for functions that is continuous but nowhere differentiable

It is well known that the Brownian motion is an example of functions that is continuous but nowhere differentiable. In addition, its distributional derivative can be interpreted in the way mentioned ...
1
vote
3answers
97 views

What is the theory of distribution that makes possible to calculate Fourier transform of the Sine function?

I am an Engineering student. Sine wave function is a power signal present from $-\infty$ to $+\infty$. I have have read that because of distribution theory,the Fourier transform of the Sine function ...
0
votes
0answers
30 views

Is it true that $|∇u(x)|^2\chi_\Omega=|\nabla (u \chi_\Omega)|^2$

Let $u\in L^\infty(\Omega)\cap H^1(\Omega)$ with $\Omega$ open, bounded and regular (as you wish) domain of $\mathbb{R}^N$. Is it true that $$ \int_\Omega |\nabla u(x)|^2 \mathrm{d} x= \int_{\mathbb{...
5
votes
1answer
61 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
1
vote
2answers
85 views

Identity for rescaled Dirac Delta, $\delta(kx)$

I´m trying to proof the following Statement. $$\delta(kx)=\frac{1}{|k|} \delta(x).$$ I already tried to proof and I got this. $$u=kx \Rightarrow x=\frac{u}{k},dx=\frac{1}{k} du \\ \int_{-\infty}^\...
1
vote
1answer
27 views

Weak solutions of Navier-Stokes are square integrable distributions?

I'm reading Lemarie-Rieusset's book Recent developments in the Navier-Stokes problem and have the following issue: he defines a weak solution to the Navier-Stokes equations on $(0,T)\times\mathbb R^d$ ...
0
votes
0answers
22 views

How do we show the inequality for $p=\infty$?

How can we show the inequality for $p=\infty$ ? Since $\overline{u} \in W^{1, \infty}(\mathbb{R}^n)$ we have that $\overline{u}'$ exists and $\overline{u}, \overline{u}'$ are essentially bounded. ...
0
votes
1answer
92 views

Construct extension of function

If $u \in W^{3,p}(\mathbb{R}^{+})$ how can we construct the catoptric extension $\overline{u}$ of $u$ in $\mathbb{R}$ (reflection) such that $\overline{u} \in W^{3,p}(\mathbb{R})$ ? EDIT: By setting $...