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
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1answer
80 views

Restrictions of distributions

Is it possible to have a distribution $u\in \mathcal{D}'(\mathbb{R})$ such that the restriction of $u$ to $(0,\infty)$ is $\frac{1}{x}$ and the restriction of $u$ to $(-\infty,0)$ is $0$? It seems to ...
0
votes
1answer
104 views

Is this integral 0?

Let $\phi \in C_0^{\infty}(\Omega)$ with $\Omega = (0,1)\times(0,1)$. Let $u\in L_2(\Omega)$ defined by $u(x,y) = 1$ for $x>y, u(x,y) = 0$ for $x\leq y$ Is there a way we can conclude ...
3
votes
0answers
158 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
1
vote
0answers
77 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Is the following operations OK (this is related to the Feynman parameter trick)? $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using ...
1
vote
1answer
75 views

Show $A$ is self-adjoint and $ f= Au$ in weak sense.

Hoi, consider $L_2(\Omega)$ with $\Omega = (0,1)\times (0,1)$ and let $u\in L_2(\Omega)$ be defined as $u(x,y) = 1$ for $x>y$ and $u(x,y) = 0$ for $x\leq y$. Let $A = \partial_x^2 - \partial^2_y$ ...
0
votes
1answer
39 views

Identifying 2 spaces of distributions

Hoi, I want to show that the space $C^{\infty}(\Omega)'$ of continuous linear functionals on $C^{\infty}(\Omega)$ can be identified with the subspace $\mathcal{E}'(\Omega)$ (distributions with ...
1
vote
1answer
50 views

Inequalities for point distribution

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
1
vote
1answer
130 views

Laplacian of the potential function

Given a potential function $$\Gamma(x)=\frac{1}{(n-2)|S^{n-1}||x|^{n-2}}$$ where $n\ge 3$ is an integer, $|S^{n-1}|$ is the volume of the sphere, $x$ is in $n$ dimension. Is it true that $$-\Delta ...
1
vote
1answer
230 views

A question on locally integrable function

Let $f$ be a locally integrable function, $f\in L_{\operatorname{loc}}^{1}(\mathbb{R}^n)$. Prove that the operator $$T_f:\phi\to\int_{\mathbb{R}^n}f(x)\phi(x)dx$$ is a distribution. (See ...
2
votes
0answers
101 views

Dirac $\delta\left( \left[\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right]^2 -k^2-m^2 \right)$

This question is related to $f(k) = 0$, but we now we consider $\delta(f(k))$, i.e. $\delta\left( \left[\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right]^2 -k^2-m^2 \right)$ We ...
1
vote
2answers
301 views

Dirac $\delta(g(x))$ with complex roots $x_i$

Hey we all know about this infamous Wikipedia page Dirac-delta composition related to the Dirac-$\delta$ in composition with a function $g(x)$. But I wonder if the roots of $g(x) = 0$ happens to be ...
3
votes
3answers
210 views

distribution with point support

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
1
vote
1answer
123 views

Dirac delta questions form Mellin transform

We know that $$ f(s)= \int_{-\infty}^{\infty}f(x)\delta (x-s) d x$$ however, is there a similar delta function so for the Mellin transform $$ f(s)=\int_{0}^{\infty}f(x)m(xs) d x$$ ? That is a ...
15
votes
2answers
1k views

Sobolev space is an algebra

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
7
votes
1answer
221 views

Delta function and integrating over level sets?

Consider the three-dimensional integral $$ \int_{\mathbb R^3} d^3x\,f(x)\delta(g(x)) $$ where $\delta$ is the dirac delta, $f,b:\mathbb R^3\to\mathbb R$ and $g(x) = 0$ on some surface $S$. Is there ...
0
votes
1answer
150 views

Derivate including Dirac's Delta(x)

Can anyone explain me this result? WolframAlpha Being the derivate of $a\cos(ax)\delta(x)$, where "$a$" is just a constant. I watched the solution step by step and its result (which is easy to ...
3
votes
0answers
50 views

Formal integration of a series of the type $-f(x-a)=\sum_{n=0}\frac{C_n}{n!}\delta^{(n)}(x-a)$

This question is inspired from an answer given to this question in the physics stackexchange, specifically the integration step going from (12) to (13). We have a distribution given as ...
1
vote
2answers
611 views

Dirac delta function

1)Prove that the dirac delta function property: $$ x\delta'(x)=-\delta(x)$$ 2)and : $$\int_{-\infty}^\infty \delta'(x)f(x)dx=-f'(0) \ $$
2
votes
1answer
158 views

Fourier transform of locally integrable function

I have a question and I don't know if it is true. Is any locally integrable function a sum of an $L_{1}$ function and another nice function (perhaps, an $L_{2}$ function). This is related to the ...
1
vote
1answer
361 views

Derivative in the sense of distributions

I have a question regarding calculating the derivative in the distribution sense of the following function: $$ f(x) = \frac{d^2 }{d x^2}|\cos|x|| $$ Maybe someone can point me in the right ...
3
votes
1answer
174 views

Distributional limit

Let $u_t(x) = t^Ne^{itx}$ for $x\geqslant 0$ and $u_t(x)=0$ elsewhere. I want to calculate the distributional limit $\lim_{t\to\infty}u_t(x)$. How would one approach such problem. I just started a ...
2
votes
1answer
96 views

Dirac function and integration by parts

I have some problems to show the following relation, apparently using integration by parts and knowing that $\phi$ denotes the density of the standard one dimensional normal distribution. $$\int ...
5
votes
1answer
2k views

what do test function mean?

I am trying to learn weak derivatives. In that we call $\mathbb{C}^{\infty}_{c}$ function as test function and we use this function in weak derivatives. I want to understand why these are called test ...
3
votes
1answer
144 views

Distributional limit of $\phi/n$

I have the following problem: Let $\phi(x) \in C_0^\infty(\mathbb{R}^n)$ satisfy $\phi \geq 0$ and $\phi(0) = 1$. Show that $\phi_n = \phi/n$ converge to $0$ in $\mathcal{D}'(\mathbb{R}^n)$. My ...
2
votes
1answer
518 views

Justification of use of delta functions/rigorous proof of Green's function for Poisson equation

I'm looking at the proofs of Helmholtz's theorem, but I'm having trouble justifying their interchange of integration and differentiation and subsequent treatment of the "integrand" as a delta ...
5
votes
3answers
267 views

Derivative of delta distribution

I'm reading Reed & Simon's book on Functional Analysis. In the chapter of locally convex spaces they say: "consider the tempered distribution $\delta'(f)=-f'(0)$, which doesn't come from a ...
1
vote
1answer
47 views

$L$ elliptic diff op $\implies$ singsupp$(u)\subseteq$singsupp$(Lu)$ for distributions $u$?

If $L:D'(\mathbb{R}^n)\to D'(\mathbb{R}^n),n\in\mathbb{N}$ is a weakly elliptic, linear differential operator with constant coefficients then for every $\Omega\subseteq\mathbb{R}^n$, and for all $u\in ...
2
votes
1answer
56 views

Example of a function $h$ such that $\overline{\partial}h$ is not a measure

Everything is in the title. Do you have an example of a function $h \in L^1(\mathbb{C})$ such that $\overline{\partial}h$ in the sense of distributions is not a (finite complex) measure ? ...
2
votes
0answers
108 views

Distributions over locally compact Abelian groups: when can they be Fourier transformed?

Pontryagin duality shows us every locally compact Abelian group---such as $\mathbb{R}^n$, $\mathbb{Z}$, the circle $\mathbb{R}/\mathbb{Z}$ or any finite Abelian group---has a Fourier transform. In the ...
3
votes
1answer
592 views

Prove that $f$ does not have a weak derivative

Consider a function $f:\mathbb{R} \rightarrow [0,1 ]$ defined by: $\begin{equation*} f(x)=\left\{ \begin{array}{rl}0 & \text{if } x\leq 0,\\ 1 & \text{if } x\geq 1, \\ 1/2 & \text{if } ...
4
votes
1answer
117 views

Distinction between “measure differential equations” and “differential equations in distributions”?

Is there a universally recognized term for ODEs considered in the sense of distributions used to describe impulsive/discontinuous processes? I noticed that some authors call such ODEs "measure ...
1
vote
1answer
483 views

differential equation with distributions

I'm stuggeling with this differential equation: $T'+T=0$ Where $T$ is distribution. I found solutions in form: $\sum_{n\in A} \frac{d^n}{dx^n}\Lambda_{c_n e^{-x}}$. This can be simplified to ...
3
votes
2answers
93 views

No distribution for associated function

Show that there is no distribution $f \in D'(\mathbb{R})$ such that \begin{equation} f(\phi)=\int e^{1/x^2}\phi(x)dx \end{equation} for every $\phi \in C_{0}^{\infty}(\mathbb{R})$ with $supp(\phi) ...
0
votes
1answer
213 views

Limit of distributions of principal value

What is the limit in $D'(\mathbb{R})$ (i.e. in the distribution sense) of \begin{equation} \lim_{t \rightarrow +\infty}\frac{e^{ixt}}{x+i0} \end{equation} where $x+i0=p.v.(\frac{1}{x})-i\pi\delta(x)$ ...
2
votes
1answer
292 views

About a property of the Dirac delta function

How can I show that there is no $u$ satisfying both (i) and (ii):$$(i) \; u \in L^p (\Bbb R^n )$$ and $$(ii) \int_{\Bbb R^n} \delta (x) \phi(x) dx= \int_{\Bbb R^n} u (x) \phi(x)dx\; ( \forall \phi ...
1
vote
1answer
772 views

Inverse function with Dirac Delta

We know that the inverse function of $ y=\log(x) $ is $ y =\exp(x) $. However, what would be the inverse of $ y=\log(x)+ \sum_{n=1}^{\infty}\delta (x-n) $? I have tried with Mathematica, and ...
10
votes
5answers
2k views

Is the Dirac Delta “Function” really a function?

I am given to understand that the Dirac delta function is strictly not a function in the conventional sense and it is a "functional or a distribution". The part which I can not understand why the ...
4
votes
1answer
162 views

Distributional differential equation, somehow related to compact support distributions

I've been mulling over a problem from Friedlander's Introduction to Distribution Theory for a few days now: in Chapter 3 (on distributions with compact support), it asks to solve the differential ...
0
votes
2answers
919 views

example of a function with compact support

Can you give an example of a function which is $C^\infty (\mathbb{R})$ having support on (-1,1) such that $ \int_{-1}^1 f(x)\,dx$=1 and $ \int_{-1}^1 xf(x)\,dx$=0. Thank you.
5
votes
1answer
1k views

Fourier transform of Cauchy principal value

I try to understand the direct computation of the Fourier transform of the distribution `Cauchy principal value' $v.p \frac{1}{x}$. I don't understand the following change of order of integration: $$ ...
3
votes
2answers
462 views

approximating Dirac delta with bounded derivatives

Consider the Dirac delta distribution $\delta$ in $\mathbf{R}^d$. It is quite standard to approximate it by functions $g_n$ with $\|g_n\|_{L^1} = 1$. Is it possible to choose a sequence of test ...
4
votes
1answer
147 views

Generalizing the weak derivative

I am wondering about the weak derivative in time. We say f has a weak derivative f' if $$\int_0^T f\phi' = -\int_0^T f'\phi$$ for all $\phi \in C_0^\infty(0,T)$. This definition uses the $L^2$ inner ...
1
vote
1answer
84 views

How to prove the density result?

How to prove that $C_c^\infty(R^n)$ is dense in $C^0(R^n)$,where the topology of $C^0(R^n)$ defined as follows $u_k\rightarrow 0$ in $C^0(R^n)$ if and only if $\forall K\subset\subset R^n$,$sup_{x\in ...
5
votes
2answers
736 views

Proving the integral of the Dirac delta function is 1

Was wondering if my solution is mathematically accurate enough: The question in the book yields: Derive $$ 1=\int_{-\infty}^{\infty} \delta(x-x_i)\ dx_i $$ From $$ ...
2
votes
3answers
3k views

Heaviside step function fourier transform and principal values

I found the following answer on SE: Fourier transform of unit step? However, it is still not clear to me and maybe somebody could explain it clearer. Problem I have the following in my notes of ...
-1
votes
1answer
239 views

Fundamental solutions of PDEs

I have two questions about solving PDEs. $L$ is an linear differntial operator In the complement of the origin, the equation $LE =\delta$ reduces to $LE = 0$. Why? What can say about solutions of $ ...
-1
votes
1answer
64 views

When Dirac function is in $H^{-m}(R^n)$?

If Dirac function $\delta\in H^{-m}(R^n)$,please give the range of $m$?
9
votes
1answer
366 views

How much can a weak derivative differ from a classical one?

Let $B$ denote the unit ball in $\mathbb{R}^n$ and let $f\in C^1(B\setminus\{0\})\cap L^1(B)$. Denote with $\nabla_c f$ the classical gradient, which is defined in $B\setminus\{0\}$, and denote with ...
6
votes
1answer
304 views

Convolution between two distributions

I want to define the convolution $*$ between two distributions $S$ and $T$. For a test function $\varphi$, can I say: $$\langle S * T, \varphi \rangle \doteqdot \langle S, T*\varphi \rangle $$ where ...
3
votes
1answer
115 views

Equicontinuity and uniform boundedness for “distributions”

Exercise. (Rudin, Functional Analysis, chapter 2, pag. 53). Let us consider the space $$ \mathcal D :=\{f \in C^{\infty}(\mathbb R), \, \text{supp}f\subseteq [-1,1] \} $$ with the topology induced by ...