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).

learn more… | top users | synonyms

0
votes
1answer
79 views

Mollifiers: Nonexample?

A smooth, compactly supported, normalized, positive, etc. function is called mollifier if: ...
2
votes
1answer
36 views

Prove that $f(x) = |x|$ belongs to $D'( \mathbb{R})$

Prove that $f : \mathbb{R} \rightarrow \mathbb{R}, f(x) = |x|$ belongs to $D'(\mathbb{R})$ and find its first and second distributional derivatives, $f', f''$. To prove its linearity I used the ...
1
vote
1answer
75 views

Generalised derivative of Cantor staircase

If we consider the Cantor staircase function, let us say $f:[0,1]\to\mathbb{R}$, as a distribution, I was wondering whether there is an explicit way to express its generalised derivative as a ...
1
vote
1answer
61 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
49 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 $\| \cdot\|_{m,\epsilon}$ ...
2
votes
1answer
96 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
32 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
1
vote
0answers
118 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
71 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 ...
1
vote
1answer
32 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
31 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
167 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
66 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
70 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 ...
0
votes
1answer
24 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
46 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
42 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
89 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 ...
0
votes
1answer
34 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
109 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
77 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 ...
2
votes
0answers
84 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 ...
5
votes
0answers
158 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
37 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 ...
1
vote
0answers
42 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
38 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
50 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
64 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
57 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
51 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
71 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
76 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 $\lim_{s \to 0}\frac{1}{\pi s^2} e^{-r^2/s^2}$ ...
4
votes
1answer
83 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
72 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
27 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} ...
1
vote
1answer
58 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
62 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
1answer
40 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
1answer
66 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
80 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
27 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
39 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 ...
2
votes
0answers
168 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
3answers
795 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 $\delta(f(x,y))$? ...
1
vote
0answers
71 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 ...
3
votes
0answers
76 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 ...
0
votes
1answer
47 views

Understanding Distributional Meanings and Test Functions for PDEs

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 if this is a link to a particularly good set of ...
3
votes
1answer
37 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 ...
2
votes
0answers
62 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}} \, ...
2
votes
0answers
67 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 ...