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
16 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
0
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1answer
24 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
0
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1answer
44 views

Strange Dirac delta distribution contradiction

Consider the following integrals in variables $x,y$ over the whole $\mathbb{R}$, where $a,b\in\mathbb{R}/0$ are constants: $$\int dx \int dy ~\delta(x-a)\delta(y-b\,x)=\int dy ~\delta(y-b\,a)=1$$ In ...
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0answers
19 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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1answer
26 views

How to prove a tempered distribution always has a order?

I just learn some distribution theory. In Friedlander's book, "Introduction to the theory of distributions" (page 97), he said: The dual of $\mathscr{S}(R^n)$ (Schwartz Space) consists of ...
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0answers
20 views

Generalized functions and vector calculus theorems

To apply the divergence theorem, for example, there are conditions on your function. Your function must be a function in the ordinary sense in the first place. But in Electrodynamics, sometimes our ...
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0answers
26 views

Weak convergence implies uniform convergence in distribution?

Consider an empirical processes $\{v_T(\theta), T\geq 1 \}$ and assume it is weakly convergent to the stochastic process $\{ v(\theta); \theta \in \Theta\}$., i.e. $$ E^*(f(v_T(\theta)))\rightarrow ...
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0answers
20 views

Understanding Tempered Distributions

I'm reading about tempered distributions in Folland's Real Analysis. In particular, tempered distributions are defined as the class of linear functionals on a Schwarz space. On page 293, Folland says: ...
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0answers
33 views

Solutions to the exercises in distribution theory.

Does any of you have the solutions to the exercises in this note? I am preparing my PhD written comprehensive exam in this summer and I also need to prepare this material. So, if someone has the ...
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0answers
35 views

Variable density in the equation of motion

At a fixed point in time, consider the equation of motion $$ \nabla \cdot \boldsymbol \sigma(u) + \boldsymbol f = \rho \ddot{\boldsymbol u} \quad \text{in $\Omega \subset \mathbb R^d$} $$ for a ...
2
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1answer
42 views

Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to ...
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1answer
72 views

Relationship between convergence in the space of bump functions $\mathcal{D}(\mathbb{R}^n)$ and the Schwartz space $\mathcal{S}(\mathbb{R})^n$

Given the space of bump functions $\mathcal{D}(\mathbb{R}^n)$, smooth and with compact support, $$\mathcal{D}(\mathbb{R}^n) := \{ v \in C^\infty(\mathbb{R}^n) : \mathrm{supp}(v) \ \mathrm{is\ ...
3
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1answer
39 views

Limit of a functional

I'd like to find: $$ \lim_{\varepsilon\rightarrow 0}\frac{\varepsilon}{\varepsilon^2+x^2}\qquad \mbox{ in }\mathcal D'(\mathbb{R}) $$ And I started with the definition: $$ \left\langle ...
2
votes
1answer
51 views

the continuous functions with norm

I'm having trouble trying to understand what does means the first expression in particular the last term in it should we add $\|f\|_{\infty} \leq \infty$ or what i can't see what is his role ...
4
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2answers
98 views

Uniqueness for tempered distributional Cauchy problems

Question. Assume that $U\in C^1(\,[0, \infty)\to \mathcal{S}'(\mathbb{R}^n)\,)$ is a solution to the following tempered distributional Cauchy problem $$\tag{CP}\begin{cases} \frac{ d U}{dt} = f ...
3
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2answers
155 views

Why Dirac's Delta is not an ordinary function?

Given the following definition of the Dirac's Delta: $$\delta: \mathcal{D}(\mathbb{R}^n) \to \mathbb{R}: \varphi \mapsto \langle \delta,\varphi \rangle = \varphi(\mathbf{0})$$ where ...
3
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1answer
29 views

“$L^\infty$ in the sense of distributions”

The following is Exercise 3 of Chapter 3 in Stein & Shakarchi Book 4: Show that a bounded function $f$ on $\mathbb{R}^d$ satisfies a Lipschitz condition $$|f(x)-f(y)| \leq C|x-y| ...
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1answer
26 views

Norm on space of test functions.

What is $\nabla^{j}f(x)$ for $f:\mathbb{R}^{n}\rightarrow{\mathbb{C}}$ in this note which is just after Exercise 1? It is mentioned there that is $d^{j}$-dimensional vector but I am not able to get ...
0
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1answer
56 views

Why $\left\langle f_a,\varphi\right\rangle$ is a distribution? $\left\langle f_a,\varphi\right\rangle$ defined inside.

So I just started studying distribution theory and I am asked to show that $\left\langle f_a,\varphi\right\rangle = \int_{-\infty}^{-a} + \int_a^{\infty} \! \frac{\varphi(x)}{|x|} \, \mathrm{d}x + ...
3
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1answer
54 views

Is every “nice” topological vector space a manifold?

Say $V$ is a topological vector space. What conditions do you need to add on $V$ to make it a (topological, maybe infinite-dimensional) manifold? For instance, can we view the Schwartz class ...
3
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1answer
32 views

What does it mean for a distribution to be in $L_2$?

I am new to Sobolev space and distribution theory. So here is what I know. Distributions are linear functionals on $C_0^\infty$. Let's look at the simplest Sobolev space. $H^1(\Omega)$ is equal to the ...
2
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0answers
36 views

A catch with Dirac Delta Function

We know that $$ \int_{\mathbb{R}} f(t)\delta(t) \mathrm{d}t = f(0) $$ if $f$ is continuous. What will it be if $f$ is not continuous? For instance, what is the value of $$ \int_{\mathbb{R}} ...
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0answers
31 views

I need to integrate with $\delta$ against something that isn't a test function!

In relation to ``How does integration over $\delta^{(n)}(x)$ work?,'' I need to evaluate $\int_{-a}^{a}f(x)\delta^{(n)}(x)\, dx$. However, while my $f$ is smooth on its domain, it can't be a test ...
0
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1answer
23 views

What test function on a open interval look like?

Let $\phi$ be a function defined on an open interval $I=(a,b)$. The closure of the set of points where $\phi\ne 0$ is called the support of $\phi$. If the support of $\phi$ is a compact set, then ...
3
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1answer
36 views

Identification of $L^2$ limits with distributional convergence

I just read the thread on "too much effort" and I would like to be more specific. Is the following reasoning correct: Let $g,g_\delta\in H^1(D)$, $D$ some domain in $\mathbb{R}^n$ with the following ...
3
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1answer
57 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
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0answers
27 views

Dirac Delta function scaling question

For a Dirac delta function with simple argument $x$ we have the trivial identity: $$dx \delta(x) = dx |a|\delta(ax)$$ Now, my question is, if the argument of the delta function gets more complex, ...
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0answers
41 views

Infinite solutions of Navier-Stokes equations

Is it a known fact that Navier-Stokes equations have exactly one (possibly infinite) solution in the space of distributions?
0
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1answer
63 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: ...
1
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1answer
32 views

Derivation of a non-continuous function with distribution theory

Consider the following function: $$f(x,t) = \left\{\begin{array}{ll} 1 & x \in [0, t] \\ 0 & x \not\in [0,t] \end{array}\right.$$ What can I say about the derivative of $f$ with respect to ...
0
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1answer
58 views

Derivatives of Norms and Absolute Values (distributions)

For example we have for $x \in \mathbb{R}$, $$\frac{\partial}{\partial x}\left| x\right| = 2\Theta(x) -1 $$ and thus $$\frac{\partial^2}{\partial x^2}\left| x\right| = 2\delta(x) $$ We also have, ...
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2answers
69 views

Is it possible to mathematically explain why solids go under mollification when heated?

Well, I'm sure that many people on MSE might object that this is not a math question, however, I think that there might be a well-posed mathematical answer to this question, or at least I hope so. We ...
3
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1answer
113 views

Topologies of test functions and distributions

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my ...
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0answers
26 views

Fourier transform of a function of characteristic function of a measure

Let $\mu$ be complex measure on $\mathbb{R}^2$ ($|\mu|$ is finite measure) and $\chi$ - its characteristic function $$ \chi(x_1,x_2) = \int_{\mathbb{R}^2} d\mu(p_1,p_2) \exp(i p_1 x_1+i p_2 x_2). $$ ...
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1answer
28 views

What would be to show $(1+|\xi|^2)^{s/2}\hat{u}\in \mathcal{S}^{'}(\mathbb R^n)$ is a function?

If $s\in\mathbb R$ define the sobolev space $H^s(\mathbb R^n)$ as $$H^s(\mathbb R^n):=\{u\in \mathcal{S}^{'}(\mathbb R^n): (1+|\xi|^2)^{s/2}\hat{u}\in L^2(\mathbb R^n)\}.$$ I have a doubt concearning ...
3
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0answers
27 views

Exponential of the derivative operator on the Schwartz space?

We consider the derivative operator $\mathrm{D}$ on the space of smooth and rapidly decreasing function $\mathcal{S}$. We denote by $P_n = \frac{1}{0!} + \frac{X}{1!} + \frac{X^2}{2!} + \cdots + ...
1
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1answer
30 views

Is $L^2(0,T;H^{-1}(\Omega)) \subset \mathcal{D}^*((0,T)\times \Omega)$?

Let $\Omega \subset \mathbb{R}^n$ be a domain. Consider the space of test functions $\mathcal{D}((0,T)\times \Omega)$ and the space of distributions $\mathcal{D}^*((0,T)\times \Omega).$ Is it true ...
3
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1answer
69 views

How to build a compact support for a function

I was wondering if it is possible to build a distribution with compact support from a function. More precisely, consider a compact set $\mathbf{K}\subset\mathbb{R}^2\setminus\{0\}$, and a function ...
0
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0answers
22 views

Wave front set of a corner

Consider the distribution $u$ defi ned on $R^2$ by $$ u(x, y) = \begin{cases} 1 &\text{if } 0 < x < 1;\, 0 < y < 1, \\ 0 &\text{otherwise}.\end{cases} $$ What is the singular fibre ...
2
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1answer
54 views

Weak Laplacian of $\|x\|^\alpha$

Let $\alpha> 0$ and consider the function $\|\mathbf x\|^\alpha = (x^2 + y^2)^{\frac{\alpha}{2}}$ defined on $\mathbb R^2$. I want to compute the Laplacian $\Delta (\|\mathbf x\|^\alpha)$ in the ...
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0answers
52 views

Two possible definitions of “vector-valued distribution”

Let $X$ be a reflexive Banach space. Define $$\tag{1} \mathcal{D}^\star(0, T; X)=\left\{ u\colon \mathcal{D}(0, T)\to X\ \text{linear and continuous}\right\} $$ where the topology on the space of ...
2
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1answer
54 views

Why the tempered distribution is zero?

My question is derived from the proof of the equation $\Delta f=f$ which has no nonzero solution in $\mathscr{S}'(\mathbb{R}^n)$. The ideal to solve this equation is to use the Fourier transform. By ...
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vote
1answer
23 views

How to compute the limit of a distribution

I would like to know if the following sequence : $ T_n = \displaystyle \sum_{k=1}^n \dfrac{1}{k^{2}} \delta_{\frac{1}{k}} $ converge in $ \mathcal{D} ' ( \mathbb{R} ) $. If it's converge in $ ...
2
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0answers
100 views

Problem on the integral representation of a tempered distribution

Suppose $\mathscr{S}(\mathbb{R^n})$ is the space of Schwartz functions, in which the seminorms have the form $$\left \| \varphi \right \|_{m}=\underset{\underset{x\in \mathbb{R}^{n}}{|\alpha|\leq m ...
0
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0answers
35 views

P.D.E's - Distribution

If we have $$\int_{\Omega} (u_m-u_n) \Delta \varphi \,dx=\int_{\Omega} |u_m-u_n|\,dx$$ for $\Delta\varphi = \operatorname{sgn}(u_m-u_n)$. Now, if I need to choose a test function $\varphi$ such that ...
2
votes
1answer
19 views

Support of the limit of a convergent sequence of distributions

I read that if $u_n \to u \in \mathcal{D}'(X)$, then $$ \text{supp} \, u \subset \bigcap_{n \geq 1} \bigcup_{m \geq n} \text{supp} \, u_m. $$ However, the proof given shows that $$ \text{supp} \, u ...
2
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2answers
24 views

Can we conclude that a distribution is a $L^2$ function by testing with $L^2$?

Let $T\colon \mathcal{D}\to\mathbb{R}$ be a distribution. Does $|T(f)|\leq\|f\|_2 \forall f\in\mathcal{D}$ imply $T=T_g$ for some $g\in L^2$? What if $T$ is tempered?
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2answers
68 views

Derivatives of $|x|$

I wanted to calculate the first and second derivatives of the function $f(x)=|x|$, in order to verify that: $$ f'(x)=\frac{|x|}{x} $$ and $$ f''(x)=2\delta(x). $$ Can you help me?
4
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1answer
76 views

Prove that the weak$^*$ topology on the space of tempered distributions is not 1st countable

Please, help me with a proof of this (apparently) known fact whose proof is out of my reach, even though I spent a considerable amount of time looking it up: The weak$^*$ topology on the space of ...
0
votes
1answer
69 views

distribution solution to xT = 0 in Schwartz space

I try to understand the Poisson summation formula from the perspective of distribution theory. However, I got stuck at a problem on the way, namely proving that the distribution solution to $xT = 0$ ...