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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1answer
34 views

What is the divergence $\operatorname{div}u$ of a $L^2(\Omega)^d$ function $u$?

Let $\Omega\subseteq\mathbb R^d$ be open. In the book Navier-Stokes Equations - Theory and Numerical Analysis by Roger Temam the author is using the divergence $\operatorname{div}u$ of a $L^2(\Omega)^...
0
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1answer
30 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 $\mathcal D(\Omega)'$ ...
2
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1answer
20 views

Show that $T$ is independent of $\epsilon$

Define $$T: C_{0}^{\infty}(\mathbb{R})\to \mathbb{R}$$ by $$T(\phi)=\int_{-\infty}^{\epsilon}\frac{\phi(x)}{x}+\int_{-\epsilon}^{\epsilon}\frac{\phi(x)-\phi(0)}{x}+\int_{\epsilon}^{\infty}\frac{\phi(x)...
2
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2answers
52 views

Limit of $\frac{1}{x+i y}$ for $y \rightarrow 0$ and distributional relations

So I know for $y \rightarrow 0$ I have the following (distrubutional) relation: $\frac{1}{x+i y} = \frac{x}{x^2+y^2} - i \frac{y}{x^2+y^2} = P(1/x) - i \pi \delta(x) $ where in the last expression ...
1
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2answers
324 views

cutoff function vs mollifiers

Q1. What are cutoff function? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions? And how they differ from mollifiers I did check ...
1
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1answer
37 views

Verifying equivalent definitions of continuity for distributions

For an arbitrary vector space $V$ over $\mathbb{F}$, consider continuous linear maps $f: V \to \mathbb{F}$ where continuity is defined as sequential continuity, i.e. if $\phi_j \to \phi$ in $V$ then $...
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0answers
27 views

Convolution of delta-ish functions

I would like to compute the convolution of a function with itself, where the function is $f(x) = \frac{\delta(x)}{x}$. When there is a shift in the delta function it is easy to compute, but this one ...
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0answers
92 views

Characterize a set of functions

While computing matrix elements of the evolution operator in Quantum Field Theory for the harmonic oscillator using the path integral formalism, I came across the assumption that all physically ...
1
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1answer
52 views

prove $\frac{n e^{-n^2 x^2}}{\sqrt{\pi }}$ converges to $\delta(x)$

How can show as $n$ goes toward infinity the sequence converges to $\delta(x)$ my problem is I don't know how to show this.
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0answers
36 views

Silly question about sampling a function (distribution vs integral representation)

I have a stupid question... if i want to sample a continuous function i can use the Dirac delta distribution $$f(x_0) = \int_{-\infty}^{+\infty} \delta(x - x_0) f(x)dx,$$ which indeed involves the ...
0
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1answer
32 views

Doubts on distributional derivative: null derivative and non negative derivative

suppose we are in $\mathbb R^n$ and fix $E\subset \mathbb R^n$ and let $\chi_E$ be the indicator function of the set $E$. I'm reading an article that says that a vector field $X$ on $\mathbb R^n$, ...
3
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1answer
58 views

Traveling delta function $\delta(x - ct)$ as a distributional solution of the wave equation

I'm trying to show that a delta function $\delta(x - ct)$ is a distributional solution of the PDE $$ D_{(0,2)}u(x, t) = c^2 D_{(2,0)}u(x,t). $$ Here $D_{(i,j)}$ means $i$-th partial differentiation on ...
3
votes
2answers
71 views

Dirac Delta function at a point

From my understanding of the Dirac Delta function, it is infinitely thin and has a value of infinity at only a particular point. I also learned that $$\int_{-\infty}^{\infty} \delta(x-a) dx = 1$$ What ...
0
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1answer
81 views

Integral definition of delta function and Kronecker symbol

I know the following two definitions for the delta function and Kronecker delta, respectively: (1) $\int_{-\infty}^{\infty}\frac{e^{iwt}}{2\pi}\mathrm{d}t = \delta(w)$ (2) $\int_{-\pi}^{\pi}\frac{e^{...
5
votes
1answer
61 views

$5$ questions on the definition of the Gelfand triple

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb F\in\left\{\mathbb R,\mathbb C\right\}$, $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\...
2
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1answer
50 views

Push forward of the Lebesgue measure is the Haar measure of the Carnot group

I have the following problem. I have a Carnot group $(\mathbb G,*)$ which is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ is stratified as $\mathfrak g= V_1\oplus\dots\...
1
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1answer
33 views

Fourier transform of a distribution null on [-1,1]

Here is an interesting problem : Let $f \in \mathcal{C}^0 ( \mathbb{R})$ bounded, and $T_f \in \mathcal{S}'(\mathbb{R})$ defined by $\displaystyle \langle T_f, \phi \rangle = \int_{\mathbb{R}} f(...
2
votes
1answer
68 views

What is meant by this notation for a sum of distributions

$\newcommand{ \reals }{ \mathbb{R}} $ $\newcommand{ \distfns }{ \mathcal{D}'(\reals) } $ $\newcommand{ \testfns }{ \mathcal{D}( \reals ) } $ $\newcommand{ \ints }{ \mathbb{Z}} $ $\newcommand{ \x }{ \...
5
votes
2answers
119 views

What is the Fourier transform of $\exp(2 \pi i / x)$?

The Fourier transform of $e^{2 \pi i / x}$ makes sense as a distribution, I believe. Does it have a nice expression in terms of functions and well-known distributions (e.g. Dirac delta)?
3
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2answers
82 views

Laplacian of a distribution

Here is a small result annoying me : Let $u$ be a distribution ($u \in \mathcal{D}'(\mathbb{R}^n)$) such that $\Delta u$ is continuous. Then $u$ is continuous. I am not able to prove this, and ...
0
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1answer
37 views

The vector-valued distribution of compact support

Let $\mathcal{H}$ be infinite dimensional Hilbert space and $D(\mathbb{R}^n)$ be the space of smooth complex functions of compact support. Consider the distribution $T: D(\mathbb{R}^n) \to \mathcal{H}...
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1answer
121 views

I want to prove $f\notin W^{1,1}(\mathbb{R},\gamma_{1})$

Let $\gamma_{1}=\mathscr{N}(0,I_{1})$ in $\mathbb{R}$ be the standard Gaussian measure. Consider the sequence $(f_{n})_{n\in\mathbb{N}}\in C_{b}^{1}(\mathbb{R})$ defined by $$f_{n}(x)=\begin{cases} 0,...
1
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1answer
69 views

Does this function define a distribution? $f(x) = \sum_{n \in \mathbb{N}} e^{ -(x-n)^2 }$

Is this a continuous linear form on $\mathcal{D}(\mathbb{R})$ ($C^\infty$ with compact support) such that for any sequence $\varphi_n\to0 \Rightarrow \langle T, \varphi_n \rangle \to 0$. $\int_\...
1
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1answer
46 views

Prove that a distribution has its primitive distribution.

Given a distribution $F$ in $\mathbb{R}$,prove that there exists a distribution $F_1$ such that $$\frac{d}{dx}F_1=F$$,and it is unique up to an additive constant.
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0answers
34 views

Conditionally convergent distributions?

The notion of conditional convergence can be extended to integrals. Can it also be extended to distributions - specifically for tempered distributions? The motivation behind this question comes from ...
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0answers
15 views

Fourier transforms of some homogeneous functions

In $2D$, what is the Fourier transform (in the sense of distributions) of functions of the form $x_i/|x|^2, 1/|x|, x_i/|x|, x_i x_j /|x|^2$, and so on? Here, $i = 1,2$. They are homogeneous and ...
2
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1answer
50 views

Convergence of complex exponential in dual Schwartz space

I am trying to solve the following problem, but I am a little bit missed. Show that the functions $e^{inx}$ and $e^{-inx}$ converge to zero in S' (dual Schwartz space) as $n → ∞$. Conclude that ...
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0answers
26 views

Representation of functional/distribution through infinte series of delta distributions possible?

Sorry, I am an engineer and this surpasses completely my math education. If I am not precise, please comment on open/unclear assumptions, and I will do my best in correcting my mistakes. QUESTION: is ...
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2answers
107 views

Support of a distribution, what does it mean?

In my course notes the support of a distribution (continous lineair functional) is defined as follows: Definitions First it defines something like open annihilation sets: An open annihilation ...
0
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1answer
79 views

First and second derivatives of max function

I have two functions $f(x)=(x-K)_+$ and $g(x) = \max\{x,K\}, x \geq 0, K = const \geq 0$. I was told that $$f'(x) = \mathbb{I}_{[K,+\infty)}(x)$$ and $$f''(x)=\delta_K(x),$$ because $(\mathbb{I}_{[...
2
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0answers
40 views

Fourier transform of distribution solution

Let $f(x)=2$ for all $x$. What is the Fourier transform of $f$? This is my solution but there are some steps I don't fully understand, I took it from an example just to get through the rest of the ...
1
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2answers
43 views

Radial distributions

here is a theorem I am not able to solve, it is about distributions (Schwartz). Let $\Omega = \mathbb{R}^2 \setminus \{0\}$. Show that for all $S \in \mathcal{D}'(\mathbb{R}_+^*)$ there exists a ...
2
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1answer
38 views

A simple calculation about distribution in the plane $\mathbb{R}^2$.

Let $\Omega = \lbrace (t,x)\in\mathbb{R}^2\mid t>|x| \rbrace$ and $T\in\mathcal{D}'(\mathbb{R}^2)$ defined by $T=\partial_t\mathbb{1}_\Omega - \partial_x\mathbb{1}_\Omega$ Prove that $$<T,...
3
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0answers
53 views

Proving an identity of the composition of the delta distribution with a differentiable function

Given a differentiable function $f$, some $x_j$ ($j \in \{1, ..., n\}$) such that $f(x_j) = 0$ $\forall j$ and $f'(x_j) \ne 0$ $\forall j$, and the following definition of the composition of a ...
1
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1answer
55 views

reference for periodic distributions $\mathcal{D}(\mathbb T).$

I am looking for some introduction to theory of periodic distributions $\mathcal{D}(\mathbb T).$ Would you please suggest some reference book? [ I am familiar with distibutions on $\mathbb R$. ...
2
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0answers
149 views

Prove that a homogeneous distribution is tempered.

Suppose $F$ is a homogeneous distribution of degree $λ$.Prove that $F$ is tempered,i.e.$F$ is continuous in the Schwartz space $\mathcal{S}$. It seems that it's an easy result in distribution theory,...
1
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0answers
35 views

Distributional derivative $Xu=0$ then $u$ is constant over the flow lines of $X$

Suppose we have a Lie group on $\mathbb R^n$, something like $(\mathbb R^n, \cdot)$, where with $\cdot$ we denote the group law on $\mathbb R^n$. Call $\mathfrak g$ the Lie algebra of $(\mathbb R^n, \...
2
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0answers
98 views

A sufficient and necessary condition for a distribution to be tempered.

Show that the distribution $F$ is tempered if and only if there is an integer $N$ and a constant $A$,so that for all $R\geq 1$, $$F(\varphi)\leq AR^N sup_{|x|\leq R,0\leq |\alpha|\leq N}|\partial_x^\...
0
votes
1answer
63 views

Are the three statements the same?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz function on $\mathbb{R}^n$. Consider two statements which have the same proof. $$ f\in \mathcal{S}(\mathbb{R}^n)\,\,\Longrightarrow\,\,f\in L^p(\mathbb{...
6
votes
1answer
102 views

The Schwartz function and the sobolev space $W^{2,p}$

How do you prove the Schwartz functions in $\mathbb{R}^n$ are dense in the space $W^{2,p}(\mathbb{R}^n)?$ Terrence tao has a version of the proof of The space $C_c^{\infty}(\mathbb{R}^d)$ of test ...
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2answers
89 views

If $f,g_j\in\mathcal{C}(U)$ and $\frac{\partial f}{\partial x_j}=g_j$ weakly $\Rightarrow$ $f\in\mathcal{C}^1(U)$

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f,g\in \mathcal{C}(U)$. If $$\frac{\partial f}{\partial x_j}=g$$ for some $j$ $(j=1,\ldots,n)$ in the sense of distributions, how to prove that $$\...
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0answers
50 views

an iff proof on the existence of weak derivative

I have trouble understanding the following proposition. Proposition $f,g\in L_{\text{loc}}^1(\Omega)$. Then $g=D^{\alpha}f$ iff. there exists $f_m\in C^{\infty}(\Omega)$ such that $f_m\to f$ in $L_{...
0
votes
1answer
58 views

Delta function in the sense of distributions

I have a problem understanding the meaning of the delta-function on the sense of distributions. E.g. I have the following equation: $$\left(\frac{d}{dt} \theta(t) \right) f(t) = \delta(t) f(t)$$ ...
0
votes
1answer
109 views

Fourier transform of the identity function $f(x)=x$

Let's say you are given $\omega_f \in \mathcal{S}'(\mathbb R)$ with \begin{align*} f \colon \mathbb{R} &\to \mathbb{R}\\ x &\mapsto x, \end{align*} and the definition of Fourier transform \...
0
votes
1answer
16 views

How to determine a function from its corresponding distribution?

If we have a function $\phi(x)$ we can determine the corresponding distribution $\phi^D$ such that: $$\forall f:L_{\phi^D}(f)=\langle\phi^D|f\rangle=\int_\mathbb{R}\phi(x) f(x) dx$$ as long as $f$ ...
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0answers
56 views

Solutions of an EDO in tempered distribution space being smooth out of the origin.

For $a\in\mathbb{C}$, let us consider the following differential equation in $\mathcal{S}'(\mathbb{R})$, the set of tempered distributions on $\mathbb{R}$: $$xT''+2T'+(a-x)T=0.$$ For $T\in\mathcal{S}'(...
2
votes
1answer
44 views

Fourier Transform of the “regular” tempered distribution of $|x|$

As the title states I am trying (without luck) to compute the Fourier Transform (in tempered distributional sense) of $|x|$, meaning ($\mathcal{S}(\mathbb{R})$ the Schwartz space): $$\widehat{|x|}[\...
1
vote
1answer
65 views

Inverse convolution of a distribution.

Notation. Let ${\mathcal{D}'}_+(\mathbb{R})$ be the set of distributions on $\mathbb{R}$ supported on $[0,+\infty[$. One easily derives the: Proposition. Let $T,S\in{\mathcal{D}'}_+(\mathbb{R})$,...
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0answers
42 views

Variational formulation curl-div equation

I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ ...
2
votes
1answer
86 views

Is $f(x)=e^x \cdot \cos(e^x)$ a tempered distribution?

Let $f(x)=e^x \cdot \cos(e^x)$. Define $$T_f(\varphi)=\int_{-\infty}^{+\infty} f(x) \cdot \varphi(x) \ .$$ I would like to know if $T_f$ defined with the formula above defines a tempered distribution ...