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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29 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 ...
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0answers
15 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 ...
2
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1answer
43 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
138 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 ...
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2answers
30 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 ...
3
votes
2answers
172 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
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0answers
11 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 ...
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0answers
30 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 ...
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1answer
25 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 ...
0
votes
1answer
33 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?
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0answers
22 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
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1answer
62 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 ...
0
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1answer
24 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
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0answers
24 views

Fourier transform of $\frac{1}{x_1^2+x_2^2+x_3^2}$ [duplicate]

How can I find Fourier transform of $$\frac{1}{x_1^2+x_2^2+x_3^2}?$$
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0answers
142 views

A generalization of the Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem on the real axis states: $$ \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{x\pm i \varepsilon}\,dx = \mp i \pi f(0) + \mathcal{P}\int_a^b \frac{f(x)}{x}\, dx $$ ...
1
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0answers
46 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 ...
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0answers
29 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 ...
1
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0answers
202 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 ...
1
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2answers
45 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
1answer
73 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 ...
0
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0answers
60 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 ...
2
votes
2answers
129 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
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2answers
52 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 ...
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1answer
78 views

Mollifiers: Nonexample?

A smooth, compactly supported, normalized, positive, etc. function is called mollifier if: ...
2
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1answer
35 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 ...
1
vote
1answer
68 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 ...
1
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1answer
61 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 ...
0
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0answers
49 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 $\| \cdot\|_{m,\epsilon}$ ...
2
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1answer
81 views

Why is the topology of compactly supported smooth function in $\mathbb R^d$ not first countable?

In other words, given a countable sequence of neighborhoods of $f(x)=0$, how to construct another open neighborhood that doesn't contain any of these neighborhoods? Thanks.
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1answer
30 views

Tempered distribution and primitive integral

$f$ is a Schwartz function on $\mathbb{R}$. Define $g(x)= \int_{-\infty}^{x} f(x)dx$. Show that $g(x)$ is a tempered distribution. Any ideas? I have no idea how to do the problem
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0answers
109 views

Convolution between Tempered distribution and schwartz function

$T$ is a tempered distribution on $\mathbb R$, $f$ is a Schwartz functions on $\mathbb R$. We define $T\ast f$ as $(T\ast f) (l)$=$T(f(-x)\ast l)$ for all $l$ Schwartz function, where the last $\ast$ ...
2
votes
1answer
69 views

Distributions (Generalized Functions)

Why is a distribution defined in terms of the inequality $$ |\langle\Gamma, \psi\rangle| \leq C \sum_{|\alpha| \leq N} \sup_{x \in S} | \partial^\alpha \psi |$$ for all $\psi \in C^\infty_c ...
1
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1answer
30 views

Distributional derivative of a hölder function

Let $f$ be a $\alpha$-Hölder function in $\mathbb{R}^n$. Question : does it have distributional derivatives in a $L^p$ space ? (modulo a suitable relationship between $\alpha$ and $n$). I know the ...
0
votes
1answer
30 views

Shifting a smooth function of compact support

Let $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function of compact support. Define $$\psi (x) := \begin{cases} \frac{\varphi (x) - \varphi (0)}{x}, & ...
0
votes
1answer
144 views

Dirac Delta function and normal distribution

I understand the Dirac Delta is the limit of a normal distribution when the variance of the normal distribution tends to 0: $$ \delta(x) = \lim_{v\to 0}\frac{e^{-x^2/2v}}{\sqrt{2\pi v}} $$ Then what ...
1
vote
1answer
64 views

If $f$ is differentiable everywhere, is $f'$ the weak derivative?

Let $f \in C^0([0,T])$ be such that $f'$ exists in the classical sense everywhere, but $f'$ may not be continuous. Is it true that $f'$ is the weak derivative of $f$ too, if it exists? I know this is ...
2
votes
1answer
67 views

How to prove a tempered distribution is in $L^p(\mathbb{R}^n)$

Given $g \in L^p(\mathbb{R}^n)$, how can I to prove that the tempered distribution $$f=\mathcal{F}^{-1}[(z-4\pi^2|x|^2)^{-1}\mathcal{F}g]$$ is in $L^{p}(\mathbb{R}^n)$ where $z \in \{u \in ...
0
votes
1answer
24 views

Weak derivative of generalized stepfunction?

Let $f$ be a function that is equal to $x$ for $t<0$ and $y$ for $t\ge0$. Can we write down the weak derivative of this function at $t=0$?
0
votes
1answer
45 views

Scaling of the delta function derivative

I'm stuck figuring out a simple scaling property for the derivative of the delta function. What relation am i missing that results in $$ \delta'(ax) = \frac{1}{a^2}\delta'(x) $$ Instead of just ...
1
vote
1answer
42 views

The non-existence of one distribution

The problem is to prove that does not exists a distribution $u$ on $\mathbb{R}$ such that $$ \langle u, \varphi \rangle = \int e^{1/x^2} \varphi(x) \, dx, \hspace{0.9cm} \varphi \in ...
3
votes
1answer
84 views

Notion of a distribution as acting on tangent spaces

I'm reading a paper that uses distributions in a way I'm not entirely comfortable with. To be precise, I'm not sure what definitions the author is working with and can't find any natural way to fill ...
0
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1answer
33 views

derivative of a step function always delta function ??

let be a piecewise continous function or 'staircase' (the function is constant everywhere but has jumps) $ F(x)$ in the sense of distribution is always true that $$ \frac{dF}{dx}= \sum_{n}\delta ...
0
votes
1answer
99 views

Dirac delta distribution & integration against locally integrable function

I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0) $, and there's no locally ...
1
vote
1answer
72 views

tempered distribution and sobolev spaces

The Schwartz space $\mathcal S(\mathbb R^d)$ is the set of all complex-valued function $f \in C^{\infty}(\mathbb R^d)$ such that $\sup_{x\in \mathbb R^d}|x^{\alpha}D^{\beta}f(x)|<\infty$ where ...
2
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0answers
78 views

Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
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0answers
155 views

Isomorphism between $C^\infty_0(B_1)$ and $\mathscr{S}(\mathbb{R}^n)$

Background: Related question I am trying to prove, that the countably-normed spaces $C^\infty_0(B_1)$ on the open unit ball (i.e. function and all derivatives vanish at the boundary) in ...
1
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1answer
37 views

How to show convergence in $\mathcal{S'}(\mathbb R^{d})$?

We put, $\mathcal{S}(\mathbb R^{d})=$ The Schwartz space and $\mathcal{S'}(\mathbb R^{d})=$ The dual of $\mathcal{S}(\mathbb R^{d})$(The space of tempered distributions). Suppose $\alpha > 1$ and ...
0
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0answers
27 views

Find $u:[0,T]\to H^2$ such that $u(0)=u_0\in H^2$ and $u_t(0)=u_1\in H^1$.

Let $u_0\in H^2$ and $u_1\in H^1$. If we define $$ \begin{align*}u:[0,T]&\longrightarrow L^2\\ t&\longmapsto u_0+\int_0^tu_1\;ds \end{align*}$$ then $u(0)=u_0$. Furthermore, the weak ...
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0answers
37 views

Distributional Representation of Perimeter in Chan-Vese

While re-implementing the classic Chan-Vese Algorithm for image segmentation, I stumbled upon the following statement, which I have problems to understand: Let $H: \mathbb{R} \to \mathbb{R}$ be the ...
1
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0answers
37 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...