Use this tag for questions about Schwartz distributions, also known as Generalised 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
20 views

Derivative of the composition of delta distribution with a differentiable function

Is there an explicit representation of what $\frac{\partial }{ \partial x} \delta(f(x,t))$ is? Here $\delta $ is the delta distribution and $f$ is an arbitrary differentiable function which is ...
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1answer
21 views

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
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2answers
67 views

What is the limit $\lim_{s \to 0}\frac{1}{\pi s^2} e^{-r^2/s^2}$

There is a typo in one of the papers I just read and instead of the known delta function limit $\lim_{s \to 0}\frac{1}{\pi s} e^{-r^2/s^2}$ it says $\lim_{s \to 0}\frac{1}{\pi s^2} e^{-r^2/s^2}$ ...
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0answers
22 views

Measure dispersion of a set of values resides in a range

I want to know what is best method to calculate (measure) the statistical dispersion of a set of values resides between a range. Scenario: My goal is to build an index. I have two methods that ...
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2answers
66 views

Example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t\notin L^\infty(0,T,H^1)$

Could someone give me an example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t$ exists (in the distributional sense), $u_t\in L^\infty(0,T,L^2)$ and $u_t\notin L^\infty(0,T,H^1)$? Thanks. ...
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1answer
21 views

Integrating a Dirac delta function with the argument dependent of a parameter

How can I handle the integral $$ \int_{t_1}^{t_2} \delta(D - x(t)) dt, $$ with $D$ a constant. I want to do a change of variables to perform the integral over $x$ but I am not sure how to proceed.
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1answer
23 views

Is it true that, $x\rho(x/t)\in H^{s}$ for $\rho\in \mathcal{D}(\mathbb R)$ and $s>3/2$?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support; and the Sobolev space $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} ...
2
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1answer
40 views

Embedding of weighted Lebesgue space on Fourier side into $C^\infty$

Given $g$, a continuous real-valued function defined in $\mathbb R^n$, $K^g:=\{u:e^g \hat{u}\in L^2(\mathbb R^n,(1/2 \pi)^nd \xi)\}$. Thus $K^g$ can be viewed as a Hilbert space with natural norm ...
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1answer
22 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
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0answers
37 views

An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
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1answer
21 views

Question on smooth compactly supported distribution

Let $u \in \mathcal{D}^{\prime}$ (i.e. $u$ is a distribution), $\phi \in C_{0}^{\infty}$ be a smooth compactly supported test function. Does $\phi u$ necessarily belong to $\mathcal{D}^{\prime}$ or to ...
2
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0answers
33 views

Speed dating/networking challenge

I am trying to organise an event with 54 participants. I want them to participate in 9 different activities at stations around a hall. Obviously this will require 9 sessions to allow the participants ...
2
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1answer
45 views

Difference between $\mathcal{E^{\prime}}$ and $\mathcal{D^{\prime}}$

What's the difference between $\mathcal{E^{\prime}}$(the space of compactly supported distributions) and $\mathcal{D^{\prime}}$ (the space of smooth compactly supported distributions)? Examples would ...
2
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1answer
45 views

Nuclearity of $\mathscr{S}$

I have big problems proving that the Schwartz Space $\mathscr{S}(\mathbb{R}^n)$ together with the topology induced by the family $$ \|\varphi\|_{p}:=\sup_{x\in \mathbb{R^n}}\sup_{|\alpha|\leq ...
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1answer
22 views

Distribution: $f_a(x)=\frac{H(x+a)-H(x-a)}{2a}$. What is its derivative with respect to the parameter $a$ and the limit as $a\to 0$.

Consider the distribution $$ f_a(x)=\frac{H(x+a)-H(x-a)}{2a}$$ Determine the $a$-derivative of this distribution $$ \left < \frac{\partial f_a}{\partial a},\phi \right> = \lim_{h\to0} ...
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0answers
22 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
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0answers
38 views

Can we characterize homogeneous Sobolev spaces by means of Sobolev embedding?

For $\alpha>0^{[1]}$, let $D^\alpha=(-\Delta)^{\frac{\alpha}{2}}$ be the operator defined on all Schwartz class functions $f$ as follows: $$ \widehat{D^\alpha f}(\xi)=\lvert \xi\rvert^\alpha ...
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2answers
34 views

How to treat Dirac delta function of two variable?

We can treat one variable delta function as $$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$ Then how do we treat two variable delta function, such as $\delta(f(x,y))$? ...
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0answers
22 views

Distribution that is not a measure

I don't know how to prove that the folowing distribution in $\mathbb{R}$ is not a measure $$T(\phi)=\phi'(0)$$ Any help will be appreciated
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0answers
43 views

What is $D\delta$ if $D$ is ordinary differential operator and $\delta$ is the Dirac distribution?

I'm reading some material about single-variable distribution theory. More specifically, I was checking some theorems of the convolution algebra $\mathcal{D}_+$, where $\mathcal{D}_+$ is the space of ...
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0answers
63 views

What is the inverse Fourier transform of $|k|^{-\alpha}$?

What is the inverse Fourier transform, $\mathcal{F}^{-1}\{|k|^{-\alpha}\}$? I am specifically interested in the case where $1<\alpha<2$. To do this, I need to compute the integral ...
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0answers
19 views

Continuous linear operator from $S'$ to $S'$

Do you know a general class of continuous linear operators from $S'$ to $S'$ ? (where $S'$ is the space of tempered distributions, dual space of the Schwartz space) Or, formally, given $X \in S'$ and ...
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0answers
48 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
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1answer
28 views

Understanding Distributional Meanings and Test Functions for PDEs

thank you for taking the time to read my question. My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even ...
3
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1answer
30 views

Limit in $S' (\mathbb{R})$

Given the sequence of distributions: $$ x^3~ \sin (nx),~~n \in (\mathbb{N}) $$ How can i find the limit for $n \rightarrow \infty$? I tried with the usual substitution $y=nx$, but it leads to ...
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0answers
20 views

Criteria to prove that a map is a tempered distribution

There is any simple sufficient condition to determine if a function is a tempered distribution? For example, given the map : $$ F \phi = \int_\epsilon^\infty \! \frac {\phi(x)}{\sqrt{x}} \, ...
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2answers
62 views

Check: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. [closed]

Part A: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. Part B: Generalize the previous problem and deduce a formula for the Fourier transform of a polynomial of degree m. ...
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0answers
59 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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1answer
43 views

The Dirac Delta Distribution $\delta_0 : D \to \mathbb R$ is not regular

I have a question on the following proof, that $\delta_0$ is not a regular distribution. We define $\delta_0$ as the linear function on test function with $$ \delta_0(\varphi) = \varphi(0) $$ for ...
2
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1answer
49 views

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that ...
2
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1answer
37 views

Distributions with support of the form $\left\lbrace x \right\rbrace$

Doing some calculations with Distributions I came up with the following theorem: THEOREM: Let $O \subseteq \mathbb{R^d}$ be an open subset and $x \in O$. Suppose $T \in \mathcal{D}'(O)$ with ...
2
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1answer
46 views

Verifying that $\lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x)$

I'd like to show that: $$ \lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x). $$
3
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1answer
37 views

When does the regularization of a function converges to the function?

Let $\theta(x)$ equal $k\exp(-\frac{1}{1-||x||} )$ if $||x||<1$, and equal 0 if $||x||\geq1.$ Here $||.||$ designates the Euclidian norm in $\mathbb{R}^{^{n}}$, and the constant $k$ is chosen such ...
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2answers
61 views

Let $\langle S, \psi \rangle=\sum_{n \in \mathbb N} \int_0^n \psi'(x)dx$. Is S a distribution?

Let $\langle S, \psi\rangle=\sum_{n \in N} \int_0^n \psi'(x)dx$. Is S a distribution? I claim that S is not a distribution. I know that if S was a distribution it would satisfy the following ...
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1answer
19 views

Example of pseudodifferential operators that smooth out the singularity of delta function

What is one example of pseudodifferential operator $P$ that smooth out the singularity of delta function, i.e. $P$ s.t. $P \delta(x) \in C^{\infty}(\mathcal{R})$?
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2answers
47 views

Show that T does not have a finite order

Part A: Show that $\langle T, \psi\rangle=\sum_{n=1}^\infty \psi^{(n)}(n)$ defines a distribution. Please check: $$|\langle T, \psi\rangle|=|\sum_{n=1}^\infty \psi^{(n)}(n)| \leq \sum_{n=1}^\infty ...
4
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1answer
73 views

Integral with Dirac Delta

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 ...
2
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1answer
49 views

Fourier transform of unit step function

It is well known that the fourier transform for unit step $U(t)$ is \begin{equation} F(U(t))=\frac{1}{j\omega}+\pi \delta(\omega) \end{equation} When I try to arrive to this expression from the ...
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1answer
33 views

Null space of $\mathrm{D_x} \mathrm{D_y}$ in $\mathcal{D}'(\mathbb{R}^2)$?

I am interested about the null space of the operator $\mathrm{D_x} \mathrm{D_y}$ on the space $\mathcal{D}'(\mathbb{R}^2)$ of generalized functions (or distributions) of Schwartz, i.e. $$\{ f \in ...
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1answer
40 views

Line integral along an implicit curve and dirac distibution

Let $\varphi : \Bbb{R}^2 \rightarrow \Bbb{R}$ defining an implicite curve $C = \{ (x,y), \varphi(x,y) = 0 \}$, and $u : \Bbb{R}^2 : \rightarrow \Bbb{R}$ Does the line integral $\int_C u(x,y)\ dC$ ...
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1answer
65 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
0
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1answer
33 views

partial derivatives of Dirac functions

I was reading my courses, and couldn't understand an exercise: The question was: simplify in $D'(\Bbb R^n)$ $\sum_{i=1}^n x_i\frac{\partial \delta}{\partial{x_i}}$ on my correction, I had written: ...
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2answers
54 views

Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...
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1answer
43 views

How to show distribution $T$ is a constant

Let $T$ be a distribution on $\mathbb{R}$. $\tau_a\phi(x)=\phi(x-a)$ and $\langle T,\tau_a\phi\rangle=\langle T,\phi\rangle$ for all $a\in \mathbb{R}$ and all test functions $\phi$. Prove that $T$ ...
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1answer
13 views

Do 3-D vectors of distributions (specifically vectors containing delta functions) have Helmholtz decompositions?

Define the function $f_i:\mathbb{R}^3\rightarrow\mathbb{R}^3$, $i\in\{1,2,3\}$, by $f_i(\boldsymbol{x})=\delta(\boldsymbol{x-x_0})\boldsymbol{e}_i$ where $\delta$ is the Dirac Delta function and ...
2
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1answer
102 views

Dirac delta distribution and measure?

Of course the Dirac delta is not a function. Despite, I think the concept of a measure is much easier than that of a distribution. Therefore, I was wondering: In what sense is the concept of a Dirac ...
3
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2answers
66 views

Topology of test functions $\mathcal{D}(\mathbb R)$

(My motivation for the following question is to understand the distribution theory) The space of test functions: $\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb ...
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1answer
50 views

Is $\langle f,g\rangle$ defined for distributions $f,g$?

Consider a standard setting for the development of the theory of distributions. Let $D(\Omega)$ be the space of test functions and $D'(\Omega)$ be the space of distributions ("generalized functions"). ...
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1answer
41 views

Prove or disprove: $e^{-nG(x)}$, normalized, is an approximation to the identity for $G(x)$ strictly convex

We are given the sequence of functions $$ \phi_{n} = \frac{e^{-nG(x)}}{\int_{\mathbb{R}}e^{-nG(x)}dx}$$ for a nonnegative, strictly convex function $G$ (that is, $G'' \geq c$ for some $c>0$) that ...
1
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1answer
23 views

Cauchy singular integral operator

Help on proving the following equality: $$K(-sgn)=S$$ where $K$ is the operator defined by $K(f)=F^{−1}fF$ ($F$=fourier transform, $f$=any function), sgn is the signum function and S is the Cauchy ...