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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132 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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34 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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45 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 ...
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1answer
55 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
108 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\ ...
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1answer
55 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 ...
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1answer
54 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 ...
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113 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 ...
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2answers
196 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 ...
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1answer
53 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
42 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 ...
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1answer
64 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 + ...
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1answer
115 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 ...
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1answer
44 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 ...
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1answer
78 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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67 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 ...
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1answer
34 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 ...
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1answer
43 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
70 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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64 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?
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1answer
247 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: ...
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2answers
47 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 ...
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1answer
126 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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79 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 ...
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1answer
212 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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34 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
39 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 ...
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60 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 + ...
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1answer
45 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 ...
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1answer
119 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 ...
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34 views

Wave front set of a corner

Consider the distribution $u$ defined on $\mathbb{R}^2$ by $$ u(x, y) = \begin{cases} 1 &\text{if } 0 < x < 1;\, 0 < y < 1, \\ 0 &\text{otherwise}.\end{cases} $$ What is the ...
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1answer
63 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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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 ...
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1answer
103 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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1answer
30 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 $ ...
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132 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 ...
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1answer
29 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 ...
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2answers
30 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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78 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?
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185 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 ...
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1answer
153 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 $xT = 0$ ...
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112 views

Approximate dirac delta and integration error

For a sequence of functions $g_n(x-x_o)$ approximating the Dirac delta I can write: $ \int_a^b g_n(x-x_o) f(x) dx = \int_a^b \delta(x-x_o) f(x) dx + \epsilon_n$ when $x_o \in [a,b]$. I am trying to ...
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2answers
81 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
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0answers
59 views

Convolution of two delta distributions

Show ${\int}_0^{\infty}\delta(x+z)\delta(y-z)dz =\delta(y+x)$ It seems obvious, and I don't think we need a rigorous proof for this (statistical mechanics homework) but I want a rigorous proof of ...
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1answer
131 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
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187 views

Delta function multiplied by an exponential function

I do not know if this is an ill-posed question but ... is $\delta(t)e^{-\gamma t}$ equal to $\delta(t)$? Thanks, biologist
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1answer
71 views

Applications of the theory of distributions outside of PDEs?

Are there any interesting, important or powerful mathematical applications to the Theory of Distributions besides those dealing with partial differential equations?
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107 views

A sequence of functions converging to the Dirac delta

let $g_n(x)=\frac{1}{2}n $ for $|x|<\frac{1}{n}$ and for positive integer n. Prove that $$\lim_{n \to \infty} g_n(x)=\delta(x)$$ Pretty evident after a quick sketch, but I don't know how to show ...
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1answer
70 views

Weighted Dirac comb as a tempered distribution?

I'm trying to determine when a "weighted" Dirac comb is a tempered distribution. More precisely, trying to prove: $$u=\sum_{k=1}^{\infty}c_k \delta_k\in\mathcal{S}'(\mathbb{R})\iff\exists ...
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2answers
56 views

A short question concerning the distributional solution of $xf=0$

I was reading my notes on the following result: All the $\mathcal{D}'(\mathbb{R})$ solutions to $xf =0$ are of the form $c\delta $ where $c$ is constant and $\delta$ is the dirac delta distribution ...