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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18 views

$\lim\limits_{n\to \infty} \arctan(nx)$ and set-valued limit?

Consider the sequence $a_n(x)=\dfrac{2}{\pi}\arctan(nx)$. $(a_n)$ converges pointwise to $1$ if $x>0$, $-1$ if $x<0$ and $0$ if $x=0$. It does not converge uniformly as the limit function is ...
0
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0answers
28 views

How is distribution defined on a compact set in $\mathbb{R}^n$?

I notice distribution is always defined on an open set. As is pointed out in the comment below, differentiability requires the domain to be an open set. And how can the distribution be defined on a ...
2
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0answers
30 views

$\langle u,\phi\rangle=0$ when ${\rm supp}(u)\cap{\rm supp}(\phi)^\circ=\emptyset$?

I know that $\langle u,\phi\rangle=0$ if ${\rm supp}(u)\cap{\rm supp}(\phi)=\emptyset$. For some while I asked myself if it's enough that $\phi$ vanishes on ${\rm supp}(u)$ but that's not true, as ...
1
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1answer
62 views

Dirac's delta, infinite series and integral

Why $\int_{-\infty}^x\sum_{i=1}^{+\infty} p^{i-1}\delta(\alpha-i)d\alpha = \sum_{i=1}^{+\infty}\int_{-\infty}^x p^{i-1}\delta(\alpha-i)d\alpha$ where $\delta$ is the Dirac's delta and $p \in ]0;1[$ ...
7
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1answer
74 views

Proving that a family of functions limits to the Dirac delta.

For each $\epsilon > 0$, define $f_\epsilon:\mathbb R\to \mathbb R$ as follows: \begin{align} f_\epsilon(k) = \frac{1}{\pi}\frac{\epsilon}{\epsilon^2+k^2}. \end{align} How does one rigorously ...
0
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1answer
42 views

two dimensional delta function

Is it correct to write $\delta(x,y)=\delta(x)\delta(y)$ where $\delta(x,y)$ is the delta function in two dimensions? Or are there some cases where the above fails to give the correct results when ...
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0answers
12 views

Fundamental solutions for wave equations which vanish inside characteristic cones

If the number of dimensions $n$ is odd, the flat-space wave operator $-\partial_t^2 + \nabla^2$ admits the fundamental solution $(-\sigma)^{1-n/2} \Theta (-\sigma)$, where $\sigma = - t^2 + |x|^2$. ...
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0answers
29 views

Distribution induced by a function

We are given $F(x) = |2x+1|, x \in \mathbb{R}$ How to determine whether $$[F|_{(0, \infty)}] \in \mathcal{D}'((0, \infty))$$ $$[F|_{(- \infty, 0)}] \in \mathcal{D}'((- \infty, 0))$$ are regular ...
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2answers
20 views

$\text{supp }u =\{0\}$ implies $u=c\delta_0$ in distributional sense?

Given $\text{supp } (u) =\{0\}$ where $u\in D(X)^\prime$ is a distribution and $X\subset \mathbb{R}$. Does this already imply that $u=c \delta_0$? for some constant $c$.
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0answers
18 views

Any way to rewrite/simplify $\int dp ~f(p)~\delta[c-g(p)-E(p)]$?

Im thinking of using $$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$$ but in my case $$\int dp ~f(p)~\delta[c-g(p)-E(p)]$$ the functions $f$ and $E$ are basically unknown, while $g$ is a ...
3
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2answers
52 views

What is the meaning of $1^\lor=\delta$

I tried to look up the definition of a $\lor$ and it does not seem to explain this particular usage $$1^\lor=\delta$$ This is used in a proof that inverse fourier transform of $1$ is $\delta$, but ...
0
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1answer
26 views

Inverse of laplacian operator

I recently read a paper, the author treats $$\int_{\mathbb{R}^d}f(y)\cdot \frac{1}{|x-y|^{d-2}}\,dx = (- \Delta)^{-1} f(y)$$ up to a constant in $\mathbb{R}^d$. I am not familiar with unbounded ...
0
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1answer
36 views

How do I find the distribution of the laplacian operator acting on Log |f|

Can someone give me some ideas/insight/suggestions on approaching this problem: Calculate the distribution $u(x) = \Delta \log{|\,f\,|}$ where $f$ is a meromorphic function that doesn't vanish ...
0
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1answer
60 views

What is ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$?

How exactly can I prove that ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$ equals the linear space $M$ spanned by 'simple fucntions' $\phi\in{\cal D}(\Omega_0\times\Omega_1)$ of the form ...
0
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1answer
29 views

Variation of Delta function integrated

We all know: $$\int_0^\infty \delta(y) dy = 1.$$ How about $$\int_{-\infty}^\infty y\delta(y) dy .$$ The solution of this is $0$. I have no idea how to get this. thx,
3
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1answer
58 views

Do sequences fully specify the topology of $\mathcal{D}(\Omega)$ and $\mathcal{D}'(\Omega)$?

It is well known that $\mathcal{D}(\Omega)$ and $\mathcal{D}'(\Omega)$ are not metrizable, and that a topological vector space is metrizable if and only if it is first-countable (Rudin, Thm. 1.24). ...
3
votes
1answer
43 views

Convergence of series of integrals

Let $\phi \in C^\infty(\mathbb R)$ be a function such that $\phi(x), \phi'(x) \to 0$ as $x \to \infty$. I want to show that $$\lim_{n \to \infty} \int_\mathbb R \cos(nx) \phi(x) \ dx = 0$$ Doing it ...
1
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0answers
14 views

Recognizing regular distributions

By "regular" distributions I understand those Schwartz distributions that arise from locally-integrable functions. Are there ways of telling them apart from the non-regular ones? Does the set of those ...
4
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1answer
35 views

Understanding Green's function

I have a problem in understanding the definition of Green's function which occurs when solving a Poisson equation $\Delta u=f$. Here is the definition of our lecture: Let $\Omega\subset\mathbb{R}^n$ ...
1
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1answer
21 views

Derivative of a distribution

$\DeclareMathOperator{\vp}{v.p.}$ We define $\vp \frac 1x \in \mathcal D'(\mathbb R)$ (the principal value of $\frac 1x$) as $$\left\langle \vp \frac 1x, \varphi \right \rangle = \lim_{\varepsilon ...
1
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1answer
36 views

Intiution behind the derivative of dirac delta function

Let me first begin what I mean by saying the intuition behind the " $\delta'(x)$ ". For example the smooth approximations of the delta function looks like the following: (Left:the smooth ...
1
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1answer
34 views

Pre-dual of distributions with support in a closed subset

Usually, in order to define the space of distributions $\mathcal{D}'(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$, one considers the space $\mathcal{D}(\Omega)$ of $C^\infty$-test ...
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1answer
90 views

How to prove that the Dirac delta is not a function?

I am currently taking a course on test functions and distributions and my task is to prove that the Dirac delta is not a function. Furthermore, I would also like to prove that it is continuous as a ...
3
votes
1answer
32 views

Defining a distribution

We fix the space $\mathcal{D}=\mathcal{C}^\infty_0(\mathbb{R}^n)$ as space of testfunctions. Let $(f_n)$ be a sequence of distributions with $\lim_{n\to\infty} f_n(\varphi)$ existing for all ...
4
votes
1answer
31 views

Differential Equation In Distributions

I need to find all $u \in \mathcal{D}'$ ( space of distributions) such that $ e^x (e^{-x} u ) ' = \delta_0 +1$. For any $\phi \in C_0^\infty(\mathbb{R})$ we have $\langle e^x (e^{-x} u ) ' , ...
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1answer
54 views

Fourier Transform of a Temperate Distribution

Let $f$ be a temperate distribution. Suppose that $f$ is a solution to the equation $ f'-f= \delta_0 +1 $. I want to find $ \hat{f}$... Here's what I did: Since $ f'-f= \delta_0 +1 $, then $ ...
2
votes
0answers
37 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
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0answers
23 views

Distribution annihilated by a vector field

Let $u$ be a distribution in $\mathcal{D}'(M)$ (the continuous dual of $\mathcal{D}(M) = C_0^\infty(M ; \mathbb{C})$), where $M$ is a smooth manifold. Let also $X$ be a smooth vector field on $M$, ...
3
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0answers
68 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
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0answers
33 views

Why $\mathcal{D}(\Omega)$ be a topological vector space is important?

Let $\Omega\subset\mathbb{R}^N$ be an open set and $\mathcal{D}(\Omega)$ the set of all infinitely differentiable functions with compact support on $\Omega$. In the study of PDEs, we use the ...
1
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1answer
30 views

Show that a distributional solution of $\Delta u = f u $ is smooth for smooth $f$

As in the title - I would like to show that if $f$ is a smooth ($C^{\infty}$) function then for any distribution $u$ satisfying $$ \Delta u = fu$$ in the distributional sense we have, in fact, $u ...
2
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1answer
53 views

Delta distribution as a bounded linear functional

Let $\Omega \subset \mathbb{R}^n$ be a domain with $0 \in \Omega$. For which $n \in \mathbb{N}$ is the Dirac-delta distribution a bounded linear functional on $H_{0}^{1}(\Omega)$, that is to say, an ...
1
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1answer
32 views

Is the solution of a PDE always the convolution of the Green function?

If we are given Poisson's equation in a space: $$\nabla ^2 u=F$$ The solutions (those who admit Fourier transform) are given by: $$u(x)=\int_\mathbb{R^n} G(x,y)F(y)dy$$ Where $G$ is the Green ...
2
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0answers
36 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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1answer
17 views

Product of $C^\infty$ and $\mathcal D$

On my book there is the following statement: We can define the product of a distribution $u \in \mathcal D'(\Omega)$ and a function $\psi \in C^\infty(\Omega)$ in the following way: ...
1
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1answer
30 views

Distribution, equation

I need to find all solutions in $\mathcal {D'}(\mathbb{R})$( distributions $T$ ) of this equation: $id \cdot T' = 0$ $T'(\varphi ) = - T(\varphi ')$ I've already solved a similar equation $id ...
0
votes
1answer
32 views

Find distributions whose second derivative is Dirac delta

How can I find distributions $T \in \mathcal{D'}(\mathbb{R})$ such that $T'' = \delta _0$ ? I know that $D^2T(\phi) = T(D^2 \phi)$. $\phi \in C^{\infty}(\Omega, \mathbb{R}), supp \ \phi$ is compact ...
0
votes
2answers
32 views

distributional derivative of $x^{-1/2}$.

Consider the distribution defined for $\phi \in C_{c}(\mathbb{R})$ by $$T(\phi) = \int_{-\infty}^{\infty} |x|^{-1/2} \phi(x) dx.$$ Compute its derivative $T^{\prime}(\phi)$. Attempt: I use ...
2
votes
1answer
107 views

For what parameters does a sequence converge in $S$

Let $S$ be space of rapidly decreasing functions $f\in C_0^\infty(\mathbb R^n)$, that for any multi-indices $\alpha$ and $\beta$ there is a constant $M_{\alpha,\beta}$ such that $$|x^\alpha D^\beta ...
3
votes
1answer
52 views

does the function $|\sin(x) |$ define a tempered distribution? if so compute the fourier transform

I need to check if the function $|\sin(x)|$ defines a tempered distribution and find the fourier transform of the distribution. I think it defines it because it is summable on every compact ...
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1answer
39 views

Distribution as limit of quotient

Let $u \in \mathcal{D}'(\mathbb{R})$. Show that: $$\frac{u-\tau_{x}u}{x} \to \mathcal{D}u$$ in $\mathcal{D}'(\mathbb{R})$ when $x \to 0$. Here, we define $\tau_{s}f(x)=f(x+s)$. It's an exercise ...
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2answers
31 views

Distributional derivative of $\sin { (\pi|x|/2))} \chi_{\{|x|<2\}}$

I have to find the first and the second distributional derivative of the function : $$f(x) = \begin{cases} \sin { (\pi|x|/2))} \quad & |x| <2 \\ 0 \quad & |x| \geq 2 \end{cases}$$ but ...
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3answers
81 views

Distributions defined by $C_0^\infty(\mathbb{R})$ enough to distinguish $f_1,f_2\in L^1(\mathbb{R})$?

Let $f_1,f_2$ be Lebesgue-summable functions on the real line. I was wondering whether space $C_0^\infty(\mathbb{R})$ of infinitely differentiable compactly supported functions, intended as ...
0
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0answers
22 views

How should I use Sobolev embedding to prove Schwartz representation theorem?

From Richard Melrose, From Microlocal analysis to global analysis, Chapter 1. Show that, for any $p\in \mathbb{R}$ the map $$ R_{p}:S(\mathbb{R}^{n})\ni \phi\rightarrow ...
3
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0answers
30 views

What is the topological interpretation of continuity of distributions?

I was given this definition of continuity in the distributional sense. A distribution $T$ over the space of test functions $\mathcal{D}$ is continuous if for every sequence of test functions $\{ ...
4
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1answer
71 views

Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Fourier series of a function that belongs to $L^2([0,T])$ can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space $L^2([0,T])$. Fourier transform of a ...
10
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2answers
272 views

Is this sequence bounded ? (An open problem between my schoolmates !)

Let $f$ be a smooth function (say $\mathcal{C}^{\infty}$) in its two real variables ($t$ and $T$). I consider the following sequence defined by $$A_n:=\lim_{T \to \infty} \int_{0}^{1} e^{-n t} ...
2
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0answers
50 views

What is a good way to prove the weak form of the Sobolev embedding theorem?

From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11. Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
0
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1answer
33 views

Proof on delta sequences

In a coursetext of mine there is a theorem that says: when $f(x)$ is locally integrable ($f(x)\in\text{L}^1_{\text{loc}}(\mathbb{R}^d$), $f(x)\geq0 \,\forall x$ and $\int f(x) dx=1$ then it follows ...
0
votes
0answers
28 views

Operator of Taylor series as a distribution?

I would like to prove a statement: $$T:=\sum_{k=0}^\infty a_k\partial_x^k\delta_0\not\in\mathscr{S}'(\mathbb{R}^n)$$ and, in contrast, $$T_n=\sum_{k=0}^n ...