Tagged Questions

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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0
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1answer
25 views

The topology on $C^\infty_c(\mathbb{R}^d)$ used for “distributions of compact support”

On the one hand, Eskin's book on PDEs tells me that I should be content to think of this topology as one "described" (not fully, and it's not even clear it's a topology) by the convergence of ...
0
votes
0answers
34 views

Topology on $C_{compact}^{\infty}(R)$

Want to show that the topology on $C_{\mathrm{compact}}^{\infty}(R)$, which is given by all the good semi-norms, is generated by the following collection of semi-norms $\| \cdot\|_{m,\epsilon}$ ...
0
votes
1answer
33 views

Why is the topology of compactly supported smooth function in $\mathbb R^d$ not first countable?

In other words, given a countable sequence of neighborhoods of $f(x)=0$, how to construct another open neighborhood that doesn't contain any of these neighborhoods? Thanks.
0
votes
1answer
16 views

Tempered distribution and primitive integral

$f$ is a Schwartz function on $\mathbb{R}$. Define $g(x)= \int_{-\infty}^{x} f(x)dx$. Show that $g(x)$ is a tempered distribution. Any ideas? I have no idea how to do the problem
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0answers
29 views

Convolution between Tempered distribution and schwartz function

$T$ is a tempered distribution on $\mathbb R$, $f$ is a Schwartz functions on $\mathbb R$. We define $T\ast f$ as $(T\ast f) (l)$=$T(f(-x)\ast l)$ for all $l$ Schwartz function, where the last $\ast$ ...
2
votes
1answer
45 views

Distributions (Generalized Functions)

Why is a distribution defined in terms of the inequality $$ |\langle\Gamma, \psi\rangle| \leq C \sum_{|\alpha| \leq N} \sup_{x \in S} | \partial^\alpha \psi |$$ for all $\psi \in C^\infty_c ...
0
votes
1answer
16 views

Distributional derivative of a hölder function

Let $f$ be a $\alpha$-Hölder function in $\mathbb{R}^n$. Question : does it have distributional derivatives in a $L^p$ space ? (modulo a suitable relationship between $\alpha$ and $n$). I know the ...
0
votes
1answer
24 views

Shifting a smooth function of compact support

Let $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function of compact support. Define $$\psi (x) := \begin{cases} \frac{\varphi (x) - \varphi (0)}{x}, & ...
0
votes
1answer
43 views

Dirac Delta function and normal distribution

I understand the Dirac Delta is the limit of a normal distribution when the variance of the normal distribution tends to 0: $$ \delta(x) = \lim_{v\to 0}\frac{e^{-x^2/2v}}{\sqrt{2\pi v}} $$ Then what ...
1
vote
1answer
49 views

If $f$ is differentiable everywhere, is $f'$ the weak derivative?

Let $f \in C^0([0,T])$ be such that $f'$ exists in the classical sense everywhere, but $f'$ may not be continuous. Is it true that $f'$ is the weak derivative of $f$ too, if it exists? I know this is ...
2
votes
1answer
38 views

How to prove a tempered distribution is in $L^p(\mathbb{R}^n)$

Given $g \in L^p(\mathbb{R}^n)$, how can I to prove that the tempered distribution $$f=\mathcal{F}^{-1}[(z-4\pi^2|x|^2)^{-1}\mathcal{F}g]$$ is in $L^{p}(\mathbb{R}^n)$ where $z \in \{u \in ...
1
vote
1answer
78 views

Cauchy Schwarz in an integral with distributions.

I am working with energy methods for PDE's and I have a expression of the following form: \begin{equation} \int f \phi\phi_{j} \end{equation} under the conditions that $f\in L^{\infty}, \phi\in ...
0
votes
1answer
20 views

Weak derivative of generalized stepfunction?

Let $f$ be a function that is equal to $x$ for $t<0$ and $y$ for $t\ge0$. Can we write down the weak derivative of this function at $t=0$?
0
votes
1answer
22 views

Scaling of the delta function derivative

I'm stuck figuring out a simple scaling property for the derivative of the delta function. What relation am i missing that results in $$ \delta'(ax) = \frac{1}{a^2}\delta'(x) $$ Instead of just ...
1
vote
1answer
35 views

The non-existence of one distribution

The problem is to prove that does not exists a distribution $u$ on $\mathbb{R}$ such that $$ \langle u, \varphi \rangle = \int e^{1/x^2} \varphi(x) \, dx, \hspace{0.9cm} \varphi \in ...
3
votes
1answer
54 views

Notion of a distribution as acting on tangent spaces

I'm reading a paper that uses distributions in a way I'm not entirely comfortable with. To be precise, I'm not sure what definitions the author is working with and can't find any natural way to fill ...
-1
votes
0answers
12 views

$\partial^{\alpha}v=0$ for all multi-index $\alpha$ with the connectedness of $\Omega$ implies a polynomial?

i have that $\partial^{\alpha}v=0$ for all multi-index $\alpha$ with the connectedness of $\Omega$ it follows from distribution theory that v is a polynomial of degree $\leq k$. how is it a ...
0
votes
1answer
25 views

derivative of a step function always delta function ??

let be a piecewise continous function or 'staircase' (the function is constant everywhere but has jumps) $ F(x)$ in the sense of distribution is always true that $$ \frac{dF}{dx}= \sum_{n}\delta ...
0
votes
1answer
38 views

Dirac delta distribution & integration against locally integrable function

I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0) $, and there's no locally ...
1
vote
1answer
34 views

tempered distribution and sobolev spaces

The Schwartz space $\mathcal S(\mathbb R^d)$ is the set of all complex-valued function $f \in C^{\infty}(\mathbb R^d)$ such that $\sup_{x\in \mathbb R^d}|x^{\alpha}D^{\beta}f(x)|<\infty$ where ...
1
vote
0answers
34 views

Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
0
votes
0answers
16 views

Description of distributions with support in a linear subspace

The following lemma is true: any distribution $\lambda$ on the real line with support included in $\{0\}$ can be written as $$ \lambda = \sum_{i = 0}^N a_i \partial^i(\delta_0)$$ with the $a_i$ being ...
5
votes
0answers
136 views

Isomorphism between $C^\infty_0(B_1)$ and $\mathscr{S}(\mathbb{R}^n)$

Background: Related question I am trying to prove, that the countably-normed spaces $C^\infty_0(B_1)$ on the open unit ball (i.e. function and all derivatives vanish at the boundary) in ...
1
vote
1answer
27 views

How to show convergence in $\mathcal{S'}(\mathbb R^{d})$?

We put, $\mathcal{S}(\mathbb R^{d})=$ The Schwartz space and $\mathcal{S'}(\mathbb R^{d})=$ The dual of $\mathcal{S}(\mathbb R^{d})$(The space of tempered distributions). Suppose $\alpha > 1$ and ...
0
votes
0answers
24 views

Find $u:[0,T]\to H^2$ such that $u(0)=u_0\in H^2$ and $u_t(0)=u_1\in H^1$.

Let $u_0\in H^2$ and $u_1\in H^1$. If we define $$ \begin{align*}u:[0,T]&\longrightarrow L^2\\ t&\longmapsto u_0+\int_0^tu_1\;ds \end{align*}$$ then $u(0)=u_0$. Furthermore, the weak ...
1
vote
0answers
17 views

Distributional Representation of Perimeter in Chan-Vese

While re-implementing the classic Chan-Vese Algorithm for image segmentation, I stumbled upon the following statement, which I have problems to understand: Let $H: \mathbb{R} \to \mathbb{R}$ be the ...
1
vote
0answers
33 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
0
votes
1answer
30 views

Distribution and Probability Distribution

I'm studying on the book of Kolmogorov and Fomin: "Elements of the Theory of Functions and Functional Analysis". I'm into the measure theory and I finished the Theorem of Radon-Nikodim. Now finally I ...
6
votes
1answer
54 views

Which normed space have Fatou's property?

There is tool in mathematics, more specifically in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property. Let $E$ be a normed ...
0
votes
1answer
39 views

Understanding how a differential equation is solved with distributions

This course page 12-13 (in French) is doing this : For a low-pass filter, the equation is $$RCy't()+y(t)=x(t)$$ with distributions it's written $$ (-RC\delta'+\delta)*y=x $$ (I don't understand this ...
1
vote
1answer
30 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 ...
2
votes
1answer
35 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 ...
1
vote
2answers
74 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 $\lim_{s \to 0}\frac{1}{\pi s^2} e^{-r^2/s^2}$ ...
1
vote
1answer
158 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 ...
4
votes
1answer
78 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. ...
1
vote
1answer
40 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.
1
vote
1answer
24 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
votes
1answer
44 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 ...
1
vote
1answer
26 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 ...
1
vote
0answers
39 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 ...
1
vote
1answer
24 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
votes
0answers
36 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
votes
1answer
46 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
votes
0answers
69 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 ...
0
votes
1answer
24 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} ...
4
votes
0answers
24 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 ...
1
vote
0answers
46 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 ...
0
votes
2answers
111 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 $\delta(f(x,y))$? ...
0
votes
0answers
23 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
1
vote
0answers
47 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 ...