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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2
votes
1answer
141 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
134 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
207 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
865 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
28 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
116 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 $g^{-1}(0)$ $$ ...
2
votes
0answers
373 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
56 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
73 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
332 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
vote
1answer
81 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
147 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
149 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
65 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
90 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
votes
0answers
172 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
vote
1answer
253 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
557 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
186 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
184 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
180 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
514 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
votes
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 ...
-4
votes
2answers
249 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
vote
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
votes
0answers
456 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
votes
0answers
39 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
vote
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
40 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
votes
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
51 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
votes
0answers
110 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
353 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 ...
4
votes
1answer
56 views

On the extension of distribution

Define a distribution on $(0,+\infty)$ by $$u(\varphi):=\sum_{k=1}^{\infty} {1 \over {k!}}\partial^k \varphi(1/k)$$ how can I show it cannot be extended to any distribution defined globally on ...
1
vote
0answers
91 views

Uniformly convergence of delta sequences

Let $(f_n) $ be a sequence of continuous functions $f_n: \mathbb R \rightarrow \mathbb R$ such that $$ \lim_{n\rightarrow \infty}\int_{\mathbb R} f_n(x) \phi(x) dx=\phi(0) $$ for each continuous ...
2
votes
1answer
49 views

A distribution problem

Let $\phi\in L^2(\mathbb{R}^3)$. Since $|\phi|^2\in L^1$, it has a distributional derivative. At least formally, $$ \nabla |\phi|^2 = \phi \nabla\overline{\phi} + \overline{\phi}\nabla\phi. $$ Is ...
5
votes
1answer
229 views

Computing the integral $ \lim\limits_{\epsilon\to 0} \int_{-2}^{0} \frac{e^{1/x(x+2)}}{x+1+i\epsilon} $

I got stuck when calculating of this expression $$ \lim_{\epsilon\rightarrow 0} \int_{-2}^{0} \frac{e^{\frac{1}{x(x+2)}}}{x+1+i\epsilon} $$ I will be grateful for the advice.
1
vote
0answers
31 views

an upper bound on Wave Front

Can you please help to understand how to solve this question: Let $f^{ij}(x)$ be a positive definite matrix smoothly varying with $x$ and define ...
3
votes
4answers
849 views

Proof: $ \int^{\infty}_{-\infty}\delta(t)^2 dt = 1 ??$

Let $\delta(t)$ be the Dirac-Delta function. I know that its area is 1, and amplitude is $\infty$. Then, how to prove that: $ \int^{\infty}_{-\infty}\delta(t)^2 dt = 1 ??$
0
votes
1answer
43 views

find the non constant polynomial so that P(x) δ_1' = δ_1'

This is the last one for today: I am trying to find a non-constant polynomial $P(x)$ so that the following equation is true: $$P(x) δ_1' = δ_1'$$ where $δ_1$ is the Dirac distribution supported in ...
2
votes
1answer
411 views

Is the sequence convergent in quadratic mean or in distributional sense?

This is another problem I am not sure about: Determine if the following sequence of functions on R : $$f_n(x) =\frac{\sin [x \sqrt n + \log(n)]}{(1 + nx^2)^\frac13}$$ converges in quadratic mean? ...
2
votes
1answer
131 views

Oscillating integral

I want to calculate $$ \int _0^\infty e^{-iyx}\sqrt{x(x+2)}\, dx $$ in the sense of distributions, at least for $y\ne 0$. Now, I happen to know the following integral representation for the modified ...
2
votes
1answer
55 views

A not smooth distribution

Is exist the not smooth distribution which satisfying: $\left ( D_{t}^{2}-D_{x}^{2} \right )u(x,t)=0$ I can't find at least one not smooth distribution like this... Thanks for the help!
0
votes
1answer
76 views

Show that that the weak* limit of $\exp(inx)$ is $0$

Let $f_n(x)=\exp(inx)$ I want to show that: $\hbox{w-}\lim f_n=0$ If I got a correct definition of w-lim I should calculate is follows: \begin{align*} \hbox{w-}\lim\ f_n &= \lim ...
0
votes
1answer
297 views

Convergence to $\delta$ distribution

Show that $$v_{t}(x) = (4 \pi kt)^{- \frac{1}{2}} \exp \left( -\frac{a x ^2}{4kt} \right)$$ converges to $\delta_{0}$ in $D'(\mathbb{R})$ when $t \to 0^{+}$. Asumming that: $$\int_{\mathbb{R}} ...