Use this tag for questions about distributions (or generalized 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
105 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
472 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
110 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
118 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
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2answers
159 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
145 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 ...
2
votes
1answer
610 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
112 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
68 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
110 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 ...
7
votes
0answers
348 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
120 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 $\varepsilon \to 0$ ...
2
votes
1answer
134 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
130 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
183 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
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3answers
793 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
27 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, ...
2
votes
2answers
115 views

Integral transform with Dirac delta

Let $f,g: \mathbb{R}^n \to \mathbb{R}$. Let $\delta$ denote the Dirac delta function. How can I write the integral over $\mathbb{R}^n$ (on the left hand side) as an integral over $g^{-1}(0)$ $$ ...
1
vote
0answers
329 views

Product rule for a distribution and a function

This should be simple enough; say f is a distribution and g is a function, show $(fg)'[t] = fg'[t] + f'g[t]$. I kept getting a negative sign when I was doing it myself, and looking up a solution, I ...
-1
votes
1answer
51 views

Handling Convergence for Derivative of a Distribution

Obtain the derivative of the distribution defined by $\rho [t] = \int_0^\infty \frac{t(x)}{\sqrt{x}}dx$, and express your answer in the form of an integral over $x$ of a formula that involves $t(x)$ ...
0
votes
1answer
68 views

distributional derivative in L^2

Assume I have a function $f \in L^2(R^d,\mu)$ or $f \in L^1(R^d,\mu)$. A. Now I know that it as distributional derivative, right? I call that $\partial f$. B. If I can show now that $\int ...
3
votes
1answer
318 views

Exercises about Distributions

I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ...
1
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1answer
77 views

Question about compactly supported distribuions

Let u be a distribution with compact support and let f be a Schwartz function: Is it true that the convolution of f with u is a Schwartz function?
4
votes
2answers
139 views

If $u$ and $v$ have weak derivatives,what about $uv$?

$\Omega$ is a domain in $R^n$, Let $u\in L^1_{\text{loc}}(\Omega)$. If there exists $g_i \in L^1_{\text{loc}}(\Omega)$ such that $$\int_\Omega g_i \phi \, dx=-\int_\Omega u \frac{\partial ...
2
votes
1answer
148 views

$ \frac{\partial^2 T}{\partial x\partial y} = 0 $, then $ T = ? $

Can we characterize all distributions $T \in \mathcal{D}'(\mathbb{R}^2) $ with the following property of distribution derivatives ? $$ \frac{\partial^2 T}{\partial x\partial y} = 0 $$ For functions it ...
2
votes
1answer
63 views

Does this distribution make any sense?

I met this distribution $u$ which acts like $$ \langle u, \phi \rangle = \int _{\mathbb{R}} \frac{\phi (t)}{t^n}\,dt , \qquad \phi \in C_0^\infty (\mathbb{R}) $$ where $n\ge 2$ is an integer. Is this ...
1
vote
1answer
87 views

About a metric over $C^{\infty}(\Omega)$

I need some help with this exercise: let $\Omega$ be an open subset of $\mathbb{R}^n$. We consider: $K_m=\lbrace{x\in\Omega, d(x,\mathbb{R}^n-\Omega)\geq\frac{1}{m},|x|\leq m}\rbrace$ If $\Phi\in ...
3
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0answers
160 views

Verifying the fundamental solution of Stokes flow in R^3?

Recall that Stokes flow in $\mathbb{R}^3$ is the solution $(\textbf{u}, p)$ of $$\begin{align} -\nabla p + \mu \Delta \textbf{u} &= \textbf{f} \\ \nabla \cdot \textbf{u} &= 0, \end{align} ...
1
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1answer
229 views

Asymptotic behavior of Fourier transform

Consider the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$, $f(x) = |x|^{-1}$. It is locally integrable, and its distributional Fourier transform is $F(f)(k) = g(k) = 4\pi/|k|^2$. Intuitively, the ...
4
votes
1answer
520 views

Order of distribution

Let $T$ be Schwartz distribution. Assume that the following inequality holds $T(\phi) \leq \textrm{const} ~\| \tilde{\phi}\|_1$ for any $\phi \in S(\mathbb{R})$ ($\tilde{\phi}=\mathcal{F}(\phi)$ is ...
1
vote
1answer
179 views

Identity with Dirac delta function: $\delta (x^2-a^2) = \frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$

How can I show that $\delta (x^2-a^2) = \frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$? I'm suppose to integrate it by a differentiable function and integrate, but I can't figure this one out.
2
votes
1answer
44 views

Identify the derivative of a distribution

When someone wants to identify the derivative of a distribution $T\in \mathcal{D}'(\mathbb{R})$, we usually write, for $\varphi\in\mathcal{D}(\mathbb{R})$ , $$<T',\varphi> = -<T,\varphi'> ...
3
votes
3answers
183 views

The inverse Fourier transform of $1$ is Dirac's Delta

From the definition of the Dirac delta $\delta_0$ one can infer that its Fourier transform is identically equal to $1$. But going in the other direction is not as straightforward. How can one show ...
3
votes
1answer
162 views

Schwartz kernel theorem in the case the distributions are induced by smooth functions..

how can I show that if $A:C^\infty(\mathbb T^n)\rightarrow C^\infty(\mathbb T^n)$ is a continuous linear operators then there is a unique linear and continuous operator $K_A: C^\infty(\mathbb ...
0
votes
1answer
415 views

meaning of fundamental solution

i would like to understand what is a mathematical,even physical meaning of fundamental solution,let us consider following problem from Wikipedia $Lf=sin(x)$ where $L$ is operator of second ...
2
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2answers
75 views

Solving equations of the form $y(x) f(x) =0$

When speaking with my advisor recently, we were led in the course of a physics problem to an equation of the form $$y(x) \ f(x) = 0$$ with $f(x)$ known and $y(x)$ unknown. My immediate instinct was to ...
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votes
2answers
213 views

Definition of distribution, pseudofunction, and tempered distribution

As a physicist, I do not know the difference What is a pseudofunction? How is it different from a distribution? What is a tempered distribution? If possible could you give examples and tell what ...
1
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0answers
36 views

delta functions of Riemann zeros

let be the fucntion $$ M(x)= \sum_{n= -\infty}^{\infty} \delta (x- \gamma _{n}) $$ here the sum of deltas run over the imaginary par of the Riemann zeros my first question is this in the sense of ...
1
vote
1answer
65 views

How to finish some complex integration

How to finish some integration as following below: $$\int_x^{\infty} \frac{\mathrm \beta^{\alpha+\gamma} X^{\alpha-1}(y-x)^{\gamma-1}\exp^{-\beta y}}{\Gamma(\alpha) \Gamma(\gamma)}dy\;$$ and ...
10
votes
3answers
1k views

Rigorous derivation/explanation of delta function representation?

I am interested in a derivation of the following representation for the Dirac delta function: $$\delta(x-a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i p (x-a)}dp$$ It is clear to me how the property ...
2
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0answers
417 views

Is there a particular meaning to the sum of Fourier coefficients $a_{n^2}$?

The formula $$\sum_{n=-\infty}^{+\infty} e^{-in^2x}$$ does not converge in any function space but it is perfectly valid in $\mathcal{D}'(\mathbb{R})$. When applied on a test function $\psi(x) = ...
2
votes
1answer
86 views

Correct notation when integrating Dirac distribution

I have a question regarding the correct notation when integrating the Dirac distribution $\mu$. When treating it as a measure, I can write the Lebesgue inetgral $\int_{\mathbb{R}}\mu(dx)=1.$ What if I ...
0
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0answers
37 views

On the evaluation of a distribution

In $\mathbb{R}^2\setminus\{(0,0)\}$, let $\theta(x,y)$ denote the branch of polar angle satisfying $-\pi<\theta(x,y)\leq \pi$. Since $\theta \in L^1_{loc}(\mathbb{R}^2)$, it can be regarded as a ...
1
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1answer
2k views

Proving that the delta function is symmetric

to prove that the delta function is symmetric, I need to show that $\delta(x) = \delta(-x)$ by employing a change in variables. $$\delta(x) = {1\over 2\pi}\int_{-\infty}^\infty\exp(ikx)dk\tag{1}$$ ...
2
votes
0answers
39 views

On the Fourier transform of a certain characteristic function

Consider Schwartz's distribution on $\mathbb{R}^2$. Let $$L=a\partial_x^2+b\partial_x\partial_y+c\partial_y^2$$ and $A:=\{(x,y)\in\mathbb{R}^2|y\geq |x|\}$. The problem asks if $L\chi_A=\delta$ as ...
2
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0answers
106 views

What is the set of all functions which can be used as a 'convergence factor' for a Fourier Transform?

At times, I am required to take the Fourier Transform of some function that does not decay quickly enough for the Fourier Transform to converge in the usual sense. For example, $$ ...
1
vote
1answer
49 views

On showing a distribution is a function

Consider the distributional equation $$\Delta \omega-\omega=\mu$$ Then it is easy to verify by Fourier transform that $$\omega=-\mathcal{F}^{-1}\left(\frac{1}{|\cdot|^2+1}\hat{\mu}\right)$$ is the ...
3
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0answers
106 views

Variation of Partition of Unity

We know as "Partition of unity" that follow: Let $X\subseteq \mathbb{R}^n$ be an open set, and let $K$ be a compact subset of $X$. Let $X_i$, $i=1,\ldots, m$, be open subsets of $X$ whose union ...
0
votes
1answer
24 views

Approximation of Delatadistribution

I'm trying to understand a computation in my physics script. To describe the Deltadistribution $\delta(x) $ correctly we would need the formalism of distributions, but one can also much less ...
3
votes
1answer
342 views

Sinc to delta function: error term

It is well known that $$ \lim_{L\to\infty} \frac{\sin(L x)}{x} = \delta(x) $$ in the sense of distribution. Does anybody know of the error term in the above equation ? I am interested in the leading ...