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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2
votes
1answer
43 views

distributions whose derivative is zero?

I just learned about the notion of tempered distributions $\mathcal{S}'(\mathbb{R})$. But it is unclear that if such a distribution has a 0 derivative (of course in the distribution sense) then it ...
1
vote
2answers
25 views

Distributional solution of this equation

I am having trouble finding the distributional solutions $u$ of: $x^2u = \delta$. Could somebody help? Thanks in advance
0
votes
0answers
23 views

Proving the absolute value of a smooth function is $W^{1,p}$ [duplicate]

How could one prove the following: Take $u \in C^1_c(\mathbb{R}^n)$ Then, $|u|$ is in $W^{1,p}(\mathbb{R}^n)$, $p \in [1;\infty)$. The problem is to show that the derivative in the distribution sens ...
0
votes
1answer
40 views

Show that the limit of distributions is Dirac delta

I would like to show that the following statement is true $ \lim_{a\searrow 0} \theta(x)\frac{x^{1-a}}{\Gamma(a)} = \delta(x). $ $\Gamma$ is the gamma function. The above limit is in the sense of ...
1
vote
1answer
33 views

Convergence of $f_n(x)=n^2f(nx)$ in the sense of distributionas

Let $f$ be a test function such that $\int_{-\infty}^\infty f(x)dx=0$ and $f_n(x)=n^2f(nx)$. Find the distributional limit $\lim_{n\to\infty}f_n$. How can I use the Dominated Convergence Theorem ...
1
vote
1answer
28 views

Distribution equation (x-a)T=0

I have to solve the equation $(x-a)T=0$ , T is a distribution. By definition : $(x-a)\int T(x)\varnothing (x)=0$ I know if I pose $X=x-a$ I find $XT(X)=0$ and $T(X)=\delta(X)$. But I stuck to find ...
3
votes
1answer
33 views

Seminorms in distribution theory are norms, right?

In distribution theory the seminorms that you use there $p_m( \phi) := \max_{|\alpha| \le m} \sup_{x \in \Omega} |(\partial^{\alpha}(\phi) (x)|, \phi \in C_c^{\infty}(\Omega)$. Those guys are norms ...
1
vote
1answer
24 views

Bound on the set of compactly supported distributions with support in the same compact set

Consider the set of all compactly supported distributions $v\in\mathcal{\mathcal{E}^{\prime}}(\mathbb{R}^{n})=\left(C^{\infty}\right)^{*}$ with compact support in a fixed compact set $\Omega$ . ...
1
vote
1answer
45 views

A variant of the fundamental lemma of calculus of variation

If $F$ is a distribution and its distributional derivative is equal to 0, how can I show that $F$ is (represented by) a constant function i.e. there exists a constant $c$ such that $F(\phi)=c\int\phi$ ...
5
votes
1answer
119 views

Distributional linear differential equations

What are the most general distributional solutions $u \in \mathcal{D}'(\mathbb{R})$ to $-\frac{d^n u}{dx^n} + c_{n-1}\frac{d^{n-1}u}{dx^{n-1}} + ... + c_0 u = 0$; $-x\frac{d^n u}{dx^n} + ...
1
vote
0answers
70 views

Complex distributions - what are the appropriate test functions?

In the theoretical physics literature on conformal field theory, one encounters distributional formulas like $$ \frac{1}{\pi}\partial_{\bar z}\frac{1}{z} = \delta(z), $$ where $\partial_{\bar z}$ is ...
7
votes
3answers
258 views

Distribution theory and differential equations.

How does distribution theory plays role in solving differential equations? This question might seem to be very general. I will try to explain, please bear with me. I understand, distributions make it ...
2
votes
1answer
49 views

Sign mistake in Fourier transform of $\frac{x}{1+x^2}$.

I want to calculate the distributional Fourier transform of $u(x) = \frac{x}{1+x^2}$ in one dimension in the distributional sense as $u\notin L^1$. I use the distributional definition of the Fourier ...
5
votes
2answers
42 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
3
votes
1answer
54 views

Show that the distribution is of the form $C \delta + f$

I'm trying to solve this problem: Let $ u = p.v.(1/x)$, $\phi$, $\psi \in C^{\infty}_c$. I want to show that the distribution $(\phi u )* (\psi u)$ is of the form $C \delta + f$ for some constant C ...
2
votes
1answer
151 views

Generalized Functions (Distributions) over Manifolds

What is the right way of making sense of generalized functions over manifolds? For concreteness, let me restrict my question to the dirac delta function. The article on Wikipedia on Dirac delta ...
0
votes
0answers
31 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 ...
2
votes
1answer
80 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 wondered whether it's enough that $\phi$ vanishes on ${\rm supp}(u)$ but that's not true, as ...
1
vote
1answer
73 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 $p \in ]0;1[$ ...
8
votes
1answer
102 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
votes
1answer
59 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 ...
1
vote
0answers
35 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 $\sigma = - t^2 + |x|^2$. ...
3
votes
0answers
47 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 ...
0
votes
2answers
28 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 $c$.
0
votes
0answers
23 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
votes
2answers
59 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
votes
1answer
122 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
votes
1answer
48 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
votes
1answer
66 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
votes
1answer
54 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
votes
1answer
74 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
49 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
vote
0answers
22 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
votes
1answer
78 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
vote
1answer
29 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
vote
1answer
97 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
vote
1answer
41 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 ...
1
vote
1answer
121 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
45 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
50 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 ) ' , ...
-1
votes
1answer
68 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 $ ...
3
votes
0answers
50 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 ...
3
votes
0answers
81 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 $$ ...
0
votes
0answers
54 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 ...
2
votes
1answer
52 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
votes
1answer
69 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
vote
1answer
70 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
votes
0answers
54 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 ...
0
votes
1answer
20 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
vote
1answer
40 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 ...