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).

learn more… | top users | synonyms

0
votes
1answer
33 views

Convolution with dirac delta - proof

I have dirac delta defined as $\delta(f)=f(0)$, where $f(x)$ is an arbitrary function. I have defined convolution of distribution and function as $T\ast f=T(\tilde{f}\ast\varphi)$, where ...
1
vote
0answers
43 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
0
votes
1answer
10 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
2
votes
1answer
36 views

How to evaluate a double integral with two Dirac functions?

Here I have a problem, is the solution the same if I integrate every one? part by part? $$\int_0^Te^{-(s+\mu\lambda^2 ) t} \int_0^l\left[\delta(x-R)\delta(t-tj)\varphi(x) \, dx\, dt\right]$$ I've ...
-1
votes
1answer
33 views

the norm of a linear operator

In in the demonstration of Lax-Milgramm lemma, they use a linear operator $A:V\to V$, where $V$ is a Hilbert space; My basic problem is how to prove that $$\|Au\|_V=\sup_{v\in ...
5
votes
2answers
338 views

Fractional derivatives of delta function $ \delta (x) $

How can I define the fractional derivative of the Delta function? I mean $D^{\alpha}= \frac{d^{\alpha}}{dx^{\alpha}} $ where $\alpha$ can be any real number, then if we define $D^{\alpha} \delta (x) ...
0
votes
1answer
16 views

Show that a particular sum converges and defines a distribution (example from Friedlander)

For $\phi \in C^\infty_0(\mathbb{R})$, define the quantity $\langle u, \phi \rangle$ by: $$ \langle u, \phi \rangle = \lim_{m \to \infty} \left[ \left(\sum_{k =1 }^m \phi\left(\frac{1}{k}\right) ...
1
vote
2answers
26 views

Convergence in $D'$

I was given by my professor of mathematical methods for physicist, a notion of convergence in $D'(\Omega)$, the space of distributions on $D(\Omega)$. Namely: A sequence of distributions $T_n\in\ ...
0
votes
0answers
16 views

Mistake in reasoning: $Pv\frac{1}{x} = 0$??

I'm making some mistake in my reasoning, which leads to $Pv\frac{1}{x} = 0$ (in distributional sense): $$<Pv\frac{1}{x},\phi> = lim_{\epsilon \rightarrow ...
0
votes
2answers
113 views

Non-Locally Integrable fundamental solutions

Given a Linear pde $L$, a distribution $u$ is said to be a fundamental solution if $Lu=\delta$ where $\delta$ is the Dirac delta distribution. A common example is the Newtonian potential which is the ...
2
votes
1answer
88 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
2answers
24 views

Support of polynomial distributions

Assume $u\in\mathcal{S}'(\mathbb{R}^n)$ is a tempered distribution such that $\widehat{u}$ is compactly supported and $u^k$ defines a distribution for each $k=1,\cdots,m$. Let $p_1,\cdots,p_m$ be ...
0
votes
0answers
14 views

Bump function construction with positive Fourier transform [duplicate]

Fellow math people, I am looking to construct a bump function with a positive and rapidly decaying Fourier transform. In particular, the function f should satisfy: (1) f non-negative and smooth and ...
3
votes
2answers
103 views

Topology on the space of test functions

I try to read into the theory of distributions and there is one thing which bothers me. I read that a distribution is a linear, continuous functional from the space of test functions, which, depending ...
1
vote
1answer
35 views

Verifying Distribution Equivalence for Fourier Series Expansion

In my lecture notes, given a periodic distribution $T \in (C_{per}^\infty([-\pi,\pi]^n))'$, the Fourier coefficients are defined by $$\hat T(m) = T({1 \over (2\pi)^n}e^{-i m \cdot x}),$$ for $m \in ...
0
votes
2answers
121 views

In what sense is $\int_{-\infty }^{\infty } \frac{x}{x^2+1} \, dx = \pi i$?

Suppose we want to give a meaning to the divergent integral $$I = \int_{-\infty }^{\infty } \frac{x}{x^2+1} \, dx,$$ perhaps in the sense of distributions or something (similarly to how $\int_{-\infty ...
0
votes
0answers
5 views

Generalised function equality

Can you explain in which sense, for a generalized function $$ Y^+(\phi):=\int_0^\infty e^{i\phi t}dt$$ we have $(-i\phi)Y^+(\phi)=1$ ? Here $\phi \in S^1$. I clearly get that $\int_0^\infty e^{i\phi ...
1
vote
0answers
25 views

How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
1
vote
2answers
67 views

Easy question about Bochner space

Question Suppose $u$ and $v$ are in $L^1(0,T; X)$ where $X$ is Banach. Suppose v = u' in the distributional sense. I want to show that, for $w \in X^*$, that $$\frac{d}{dt}\langle w, u \rangle = ...
2
votes
1answer
352 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
0
votes
1answer
27 views

Support of polynomial distribution

Let $P(x_1,\cdots,x_n)$ be a polynomial in $\mathbb{R}^n.$ What is supp$(\widehat{P})$ when $P$ viewed as a tempered distribution. Can supp$(\widehat{P})$ be the boundary of an sphere?
0
votes
1answer
19 views

Support of tempered distribution under exponetiation and differentiation

Suppose $u$ is a tempered distibution in $\mathbb{R}^n$. How are supp$(\widehat{u})$ and support of $\sum_{|\alpha|\leq k}\frac{\partial^{\alpha}\widehat{u^n}}{\partial x^{\alpha}}$ compared , where ...
5
votes
2answers
357 views

Square root of compactly supported C-infinity function

Given $u \in \mathcal{C}^\infty_0(\mathbb{R}^n)$, $u \geq 0$ everywhere, is $v(x) = \sqrt{u(x)}$ also in $\mathcal{C}^\infty_0$? It is clear that the only problematic points are the boundary of the ...
4
votes
1answer
73 views

A catch with Dirac Delta Function

We know that $$ \int_{\mathbb{R}} f(t)\delta(t) \mathrm{d}t = f(0) $$ if $f$ is continuous. What will it be if $f$ is not continuous? For instance, what is the value of $$ \int_{\mathbb{R}} ...
0
votes
0answers
31 views

The space of distribution $H^{-1}$

Let's suppose to have a function $u$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$ with $\partial_t u\in L^\infty(0,T;H^{-1}(\mathbb{R}^n))$. So $\partial_t u$ is a linear and continuous functional ...
2
votes
2answers
121 views

How to integrate $I(k)=\frac{2\pi}{i k^2}\int_0^{\infty}\left(e^{-ir }-e^{ir }\right)dr$

I heard that you can integrate $$\begin{align}I(k)=\frac{2\pi}{i k^2}\int_0^{\infty}\left(e^{-ir }-e^{ir }\right)dr \end{align}$$ in the sense of tempered distribution. Unfortunately, I am only ...
1
vote
0answers
18 views

$L^{2}$ convergence and converence of distribution

Suppose that $f_{n}(x)$ are a sequence of $L^{2}$ functions which converge to a function $f(x)$ in the $L^{2}$ sense. Show that it also converges weakly in the sense of distributions, ie for any test ...
2
votes
1answer
40 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
1
vote
0answers
179 views

computing how distributional derivatives behave under coordinate transformations

Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth ($C^\infty$) boundary. For $k > 1$, assume that $u \in C^{k-1}(\Omega)$ such that its order $k$ derivatives exist in $\Omega$ and ...
0
votes
1answer
22 views

convergences in $\mathcal {S'}$

strong textLet $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1$ if $|\xi|\leq 1.$ My Question is: ...
0
votes
0answers
20 views
0
votes
0answers
29 views

A generalization of the Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem on the real axis states: $$ \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{x\pm i \varepsilon}\,dx = \mp i \pi f(0) + \mathcal{P}\int_a^b \frac{f(x)}{x}\, dx $$ ...
0
votes
0answers
21 views

Dirac delta composed with a function and implicit equation for the roots

I'm considering an expression of the form $$\int_{-\infty}^\infty dx G(x) \delta(x^2-f(x)^2) $$ where $G$ and $f$ are two unrelated smooth functions of $x$. Now I know that when $f$ is a positive ...
1
vote
0answers
22 views

How to decompose tempered distribution by entire analytic functions?

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1$ if $|\xi|\leq 1.$ Let $j\in \mathbb N$ and ...
6
votes
5answers
9k views

Derivative of a Delta function

I know questions similar to this one have been asked, but there is a particular aspect that I'm confused about that wasn't addressed in the answers to the other ones. I'm dealing with an expression ...
0
votes
1answer
43 views

Finding Distributional Solution

In the range $0 \leq r < \infty$, find the solution of the equation $$\frac{d^{2}u}{dr^{2}} + \frac{2}{r} \frac{du}{dr} - \frac{n(n+1)}{r^{2}} u = a \delta(r-R),$$ where $n$ is an integer and ...
1
vote
2answers
29 views

Absolutly integrable functions are injective to tempered distribution?

We had a theorem that$$\mathcal{L}^1(\mathbb{R}^n)\hookrightarrow \mathscr{S}'(\mathbb{R}^n)$$ Where $\mathscr{S}'$ is the space of tempered distributions. In the proof our lecturer constructed a ...
0
votes
3answers
245 views

How to treat Dirac delta function of two variable?

We can treat one variable delta function as $$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$ Then how do we treat two variable delta function, such as $\delta(f(x,y))$? ...
0
votes
0answers
44 views

symmetric and anti symmeric distribution - sqrt function on it

I've got question for homework and I'm not sure about it, I appreciate your help. 1. assuming distribution is anti symmetric, if we apply the function sqrt on it, will we get symmetric distribution ...
1
vote
2answers
35 views

$\int_{\mathbb{R}^2} \delta(E-ax-by) x^2 dx $

I am wondering how we have to integrate $\int_{\mathbb{R}^2} \delta(E-ax^2-by^2) x^2 dxdy.$ I am not familiar with this kind of delta distribution (depending on two coordinates), so I was wondering if ...
0
votes
0answers
39 views

Apply delta function on Fourier transforms

I would like to filter out a particular frequency $\omega_o$ from the Fourier transform of a function in the form of $\left<\delta(\omega-\omega_o), \mathcal{F}(f)\right>$. If $f(t) ...
0
votes
1answer
53 views

Approximate Identity: Nonexample?

A smooth, compactly supported, normalized, positive, etc. function is called mollifier if: ...
2
votes
1answer
29 views

Prove that $f(x) = |x|$ belongs to $D'( \mathbb{R})$

Prove that $f : \mathbb{R} \rightarrow \mathbb{R}, f(x) = |x|$ belongs to $D'(\mathbb{R})$ and find its first and second distributional derivatives, $f', f''$. To prove its linearity I used the ...
5
votes
1answer
155 views

Compatibility of pointwise and distributional convergence

This has probably been asked before but I couldn't find it. Let $\Omega$ be an open subset of $\mathbb{R}^n$ and let $u_k,\, u \in L^1_{\mathrm{loc}}(\Omega)$ and $v\in \mathscr{D}'(\Omega)$ (the ...
2
votes
4answers
366 views

What is the appropriate topology on $C_c^\infty (\mathbb{R}^d)$?

Let $\{ U_k:k\in \mathbb{N}\}$ be an increasing sequence of open subsets of $\mathbb{R}^d$ whose union is $\mathbb{R}^d$ and such that each $K_k:=\overline{U_k}$ is compact and $K_k\subseteq U_{k+1}$. ...
0
votes
1answer
42 views

The topology on $C^\infty_c(\mathbb{R}^d)$ used for “distributions of compact support”

On the one hand, Eskin's book on PDEs tells me that I should be content to think of this topology as one "described" (not fully, and it's not even clear it's a topology) by the convergence of ...
1
vote
1answer
42 views

Generalised derivative of Cantor staircase

If we consider the Cantor staircase function, let us say $f:[0,1]\to\mathbb{R}$, as a distribution, I was wondering whether there is an explicit way to express its generalised derivative as a ...
-4
votes
2answers
102 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 ...
0
votes
0answers
41 views

Topology on $C_{compact}^{\infty}(R)$

Want to show that the topology on $C_{\mathrm{compact}}^{\infty}(R)$, which is given by all the good semi-norms, is generated by the following collection of semi-norms $\| \cdot\|_{m,\epsilon}$ ...
1
vote
1answer
335 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 ...