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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7
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0answers
96 views

In the space of distributions, how big is the subspace of functions?

I'm teaching Distribution theory and many of my students still believes that there is only one kind of distribution : the distribution that can be identified to a $L^1_{\text{loc}}$ function. And I ...
0
votes
1answer
33 views

Differentiability of smooth functions

The space is from Schwartz space $S$ or the space of all smooth functions of compact support $D$. Can you say anything about how many times the test function is differentiable from this? I think ...
0
votes
2answers
155 views

Discontinuity of Dirac Delta distribution

I know that the following holds for Step function, but not sure if it holds for the distribution too. Does the following hold for the Dirac delta distribution too? $\delta$ is a linear functional ...
2
votes
0answers
36 views

Multiplying and dividing distributions by non-$C^\infty$ functions.

It's quite easy to see that we can multiply distributions by any $\mathcal C^\infty $ functions. Moreover, if the distribution $T$ is of order $k$, then we can mupliply it by a $\mathcal C^k$ ...
3
votes
0answers
63 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
8
votes
0answers
83 views

Delta distributions with nonlinear arguments

I am confused by the use of nonlinear arguments with the Dirac $\delta$ distribution that I am encountering in the literature. This looks like a widespread use, but for concreteness let us focus on a ...
1
vote
2answers
105 views

About a Fourier transform of a non- integrable function.

I'm trying to obtain the Fourier transform of the following function: $$F(x)=\frac{x}{1+x^2}$$ I have tried using Residue Theorem, but i think it can't be applied because the difference between the ...
1
vote
1answer
77 views

About convolution and Fourier transform

I have some doubts with this question: I we have $f,g\in\cal{S}$ (where $\cal{S}$ is the Schartz space) with $f\ast g=0$, Can we deduce that $f=0$ or $g=0$? What I did is apply Fourier transform, ...
1
vote
1answer
54 views

About the k-th derivative of the Delta function

I need some help to compute the k-th derivative of the Dirac's Delta function, $\delta_0^{(k)}$. I know its Fourier transform is $TF(\delta_0^{(k)})(y)=(iy)^{k}$( I don't know if this could be ...
1
vote
1answer
129 views

About the Fourier transform of the sign function

I'm trying to calculate the Fourier transform of the function $f(x):=sign(x)$. I have read some texts where this is solved approximating the function $f$ by other functions, $f_a$, defined as follows ...
0
votes
2answers
112 views

Dirac's delta definition

Wikipedia gives the following definition to Dirac's delta: $$\delta(x-\alpha)=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ip(x-\alpha)} dp $$ but solving the integral we get: $$\delta(x-\alpha) = ...
7
votes
4answers
201 views

Iterated Limits Schizophrenia

Consider the functions $g_n(x)$, with $n\in\mathbb{N}$, $n \ge 1$ and $x\in\mathbb{R}$, defined as follows: $$ g_n(x) = \begin{cases} 2n^2x & \text{if }0 \le x < 1/(2n) \\ ...
3
votes
1answer
62 views

What´s wrong in this computation of $\Delta(r^{-1})$ as a distribution?

maybe this is an idiot question, but I could not figure out what´s wrong. I know how to compute $\Delta (r^{-1})$ in $\mathbb{R}^{3}$ putting a ball with center in $0$ and then get $\Delta(r^{-1}) = ...
0
votes
1answer
54 views

Differential equation

What method should I follow if I want to solve the equation $u''-u=\delta_0+\delta_1$ in $\mathcal{D}'(\mathbb{R})$ ?? Thanks in advance!
2
votes
1answer
377 views

Fourier Transform of Dirac Comb on $\mathbb{Z}$ and $\mathbb{Z}^{d}$.

Let $f(x)=\sum_{n\in\mathbb{Z}}\delta(x-n).$ (a) Show $f$ is a tempered distribution. (b) Compute $\hat{f}$ using the convention $\int_{\mathbb{R}}f(x)e^{-ix\xi}\;dx$ convention for $\mathcal{F}$. ...
3
votes
1answer
115 views

If $f\in L^1(\mathbb{R})$ is such that $\int_{\mathbb{R}}f\phi=0$ for all continuous compactly supported $\phi$, then $f\equiv 0$.

I am wondering about a proof of the fact that If $f\in L^1(\mathbb{R})$ is such that $\int_{\mathbb{R}}f\phi=0$ for all continuous compactly supported $\phi$, then $f\equiv 0$. I am familiar with the ...
4
votes
0answers
79 views

Regularity theorem for Laplacian

Let $\Omega \subset \mathbb R^d$ be a bounded domain, $d>2$. Let $f \in C^\infty(\Omega)$. If $u \in L^2$ is a distributional solution of $\Delta u = f$ in $\Omega$ then $u \in C^\infty(\Omega)$ ...
3
votes
2answers
81 views

Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
2
votes
1answer
41 views

A distribution $u=\frac{1}{x}$

I am interested in finding a distribution $u \in \mathcal{D}'(\mathbb{R})$ such that $u=0$ on $(-\infty,0)$ and $u=\frac{1}{x}$ on $(0,\infty)$. This is exercise 1.4 in Friedlander. Hints or help ...
1
vote
0answers
104 views

Fundamental solution of wave equation in 3D

I want to ask for assistance in verifying the fundamental solution of the wave equation in $\mathbb{R}^{3}$. Here the fundamental solution is given by $$\frac{1}{2\pi}H(t)\delta(t^{2}-|x|^{2})$$which ...
0
votes
2answers
81 views

What is the closure of $ C^\infty_c(\mathbb{R}^n\setminus\{0\})$ in Sobolev $ W^{1,p} $ norm?

For $1 \leq p < \infty, n\geq 1 $ my guess of the answer was $ W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $ \overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq ...
0
votes
0answers
30 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
0
votes
1answer
100 views

Convolution of functions and measures

I need some help with this exercise. I'm not sure how to deal with it: Let $f(x)=e^{-x^2}$, $\mu$ the Lebesgue measure in $[0,1]$ and $\nu$ the Lebesgue measure in $[2,\infty)$. I have to find the ...
0
votes
1answer
53 views

Tempered fundamental solutions

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
1
vote
1answer
90 views

Convolution of distributions is not associative

I need some help with this exercise: It proposes to show that convolution of distributions is not associative: If $T=T_1$ (distribution given by f=1), $S=\delta'$, and $R=T_H$ (we denote as $H$ the ...
2
votes
2answers
133 views

How to treat $\int_{0}^{\infty} \sin(kx)dx =\frac {1}{k}$ as a distribution?

How to evaluate the following integral? $\int_{0}^{\infty} \sin(kx)dx=\frac 1 k$ The book Mathematical Physics by Butkov reads "The sequence $f_N(k)=\int_{0}^{N} \sin(kx) dx=\frac{(1-\cos ...
6
votes
1answer
136 views

What is $\Delta\frac{1}{|\mathbf{x}|^2}$, as a distribution?

$\newcommand{\x}{\mathbf{x}}$Let $\x$ denote a vector in $\mathbb{R}^3$, $|\x|$ its magintude and $\Delta=\frac{\partial^2}{\partial x 2}+\frac{\partial^2}{\partial y 2}+\frac{\partial^2}{\partial z ...
1
vote
1answer
87 views

Primitive of a distribution

I need some help with this exercise, about calculating the primitive of a distribution $T$ given by a series. Is the following: ...
1
vote
0answers
42 views

Neumann boundary condition for smooth function defined on the interior

Let $\Omega\subset\mathbb{R}^n$ be open and let $f\in C^\infty(\Omega)$ be a smooth function. What examples can one come up with that distinguish the 3 criteria below? 1: f satisfies the Neumann ...
2
votes
1answer
55 views

Does $H_0^1(\Omega)$ embed into $H_0^1(R^d)$?

Given a domain $\Omega$ in $\mathbb{R}^d$ and a function $f\in H_0^1(\Omega)$, the closure of the test functions on $\Omega$, does the extension of f by 0 to all of $\mathbb{R}^d$ necessarily lie in ...
1
vote
1answer
84 views

Distributional derivative of a characteristic function

I need some help with this exercise about distributional derivatives: If we have $N=2$, and a function $g=\chi_{C}$, where $\chi$ is the characteristic function, and $C$ is the unitary cube ...
2
votes
0answers
58 views

Fourier analysis on bounded domain?

For tempered distributions on $\mathbb{R}^n$ we can write $\widehat{\nabla f}(p)=p\hat{f}(p)$ and hence by Plancherel, we have equations like $(\nabla f,\nabla g)=(p\hat{f}(p),p\hat{g}(p))$ for ...
0
votes
1answer
64 views

A distribution that is not a Radon measure

I need help with this question: Let $N=2$, $\Omega=\mathbb{R}^2$ and $T:\cal{D}(\Omega)\to\mathbb{C}$, defined as: $<T,\phi>=\frac{\partial^2\phi}{\partial x\partial y}(0)=D^{(1,1)}\phi(0)$ I ...
2
votes
1answer
131 views

Convergence of a integral - heat Kernel and dirac delta function

Consider $\varphi \in S(R^n)$ (space of rapidly decreasing functions). Consider the heat kernel $$ K_t(x) = \displaystyle\frac{1}{{(4\pi t)}^{n/2}} \displaystyle e^{- \displaystyle\frac{|x|^2}{4t}}, ...
1
vote
1answer
86 views

Radon measure not locally integrable

I need some help with this exercise: If we have $N=2$, $\Omega=\mathbb{R}^2$ and $T:\cal{D}(\Omega)\to\mathbb{C}$, with $$\langle T,\phi\rangle=\phi(0,1)-\phi(1,0)$$ I have to show that it is a ...
-1
votes
2answers
94 views

distributional Laplacean of a function and the dirac delta distribution

Consider $S(R^2)$ the space of rapidly decreasing functions (http://en.wikipedia.org/wiki/Schwartz_space). Consider $F(x) = \displaystyle\frac{1}{2 \pi} \ln|x| , x \in R^2 - \{ 0 \}$. I want to ...
1
vote
2answers
118 views

Distribution theory problem

I need some help with this problem related with distributions: With $\cal{D}(\Omega)$ we denote de set of the functions of class $C^{\infty}$ in $\Omega$ and compact support. Let N=3. We consider ...
2
votes
1answer
68 views

Continuity condition for distributions in Rudin Functional Analysis

On page 156 of Rudin's Functional Analysis, he gives the following condition for a linear functional over the test functions $D(\Omega)$ to be continuous: A linear functional $\Lambda$ on ...
1
vote
1answer
217 views

about the derivative of dirac delta distribution

Consider the delta dirac distribution $\delta (\varphi) = \varphi (0), \varphi \in S(R^n)$ (the Schwartz space). I know that $\delta ^{'} (\varphi) = - {\varphi }^{'} (0)$. How can I prove ...
1
vote
1answer
74 views

On the composition of smooth funtions with distributions (generalized functions)

I'm trying to understand how the composition of a distribution with an infinitely differentiable function is defined and I was unable to find such a definition on the net. I read in the wikipedia ...
0
votes
1answer
52 views

Confused by a proof in Strichartz' book on Fourier Transforms

Hi I'm confused by a proof on page 53 in Strichartz book on Fourier Transforms. Specifically, in the first equation on page 53, why is it valid to interchange the action of the distribution with the ...
1
vote
1answer
75 views

Does distributional convergence imply weak convergence

let $g_k,g\in H^1(\Omega)$ (bounded domain) be given, with $g_k\to g$ in $L^2(\Omega)$. Unfortunately, I don't know whether the $g_k$ are uniformly bounded in $H^1$. I want to show that ...
5
votes
0answers
175 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
1
vote
3answers
107 views

Convergence of $\frac{1}{\sqrt{2\pi i\varepsilon}}\exp\left( -\frac{x^2}{2i\varepsilon}\right)$ as $\varepsilon \to 0$

Is it true that $\frac{1}{\sqrt{2\pi i\varepsilon}}\exp\left( -\frac{x^2}{2i\varepsilon}\right)$ converges (in some sense) to $\delta_0$, Dirac delta distribution at point $0$, as $\varepsilon \to 0$ ...
0
votes
0answers
58 views

Fourier Transform of $|x|^{\lambda}$ in $\mathbb{R}^n$

As said in the title, the problem is to prove the Fourier Transform of $|x|^{\lambda}$ in $\mathbb{R}^n$ is $|\xi|^{-\lambda-n}$, where $\lambda>-n$. The hint provided was to show that ...
2
votes
1answer
75 views

How to show a function is a test function?

How to show that $$\psi =(x^2\phi)'$$where $\phi$ is a test function, is a test function if and only if $\int_{-\infty}^{\infty} \psi dx=\int_{0}^{\infty} \psi dx=\psi(0)=0$
1
vote
1answer
85 views

Solve the following differential equaction in the sense of distribution

I have a following problem in functional analysis $$x^2\frac{du}{dx}=0$$ and I know I should solve it like this $$\langle x^2u',\phi \rangle=\langle0,\phi\rangle \Rightarrow \langle u, (x^2\phi)' ...
4
votes
1answer
92 views

Representing the dirac distribution in $H^1(\mathbb R)$ through the scalar product

Since in dimension $1$, $H^1$ is continuously embedded in $C_0$, we know that the Dirac distribution $\delta_0 \in H^1(\mathbb R)'$. Then by Riesz representation theorem we know that there exists a ...
1
vote
3answers
398 views

Integrals with Dirac delta function, $\int\delta[(x-a)(x-b)]f(x)\, dx $

I am struggling to find the result of the following integrals with dirac delta function. Why are they true? For the second one, I thought $\delta(x_1-x_2)$ must be zero?
0
votes
1answer
25 views

For every $A\in \mathcal{L}(C^\infty(\mathbb T^n))$ exists $k_A\in C^\infty(\mathbb T^n\times \mathbb T^n)$ such that..

does anyone know whether it is true that for every $A\in\mathcal{L}(C^\infty(\mathbb T^n))$ there exists $k_A\in C^\infty(\mathbb T^n\times \mathbb T^n)$ such that, $$ Af(x)=\int_{\mathbb T^n} k_A(x, ...