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
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1answer
26 views

Distribution for function

I would like a good book to study distribution or generalized functions like the "Basic idea" of that Wiki page. Is there anyone could give me some good book references in this domain? Thanks!
5
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2answers
74 views

Derivation of Dirac delta function

Is there anyone could give me a hint how to find the distributional derivative of the delta function $\delta$? I don't know how to deal with the infinite point.
1
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1answer
15 views

The convolution is in $L^1$

According to my notes: The convolution is in $L^1$. Indeed $$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) ...
0
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0answers
28 views

Why is it necessary that test functions have finite support?

For example, if $\phi(x)$ is a test function, which means smooth and with finite support the following is true: $$\lim\limits_{n->\infty} \int\limits_{-\infty}^{\infty} ...
6
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1answer
65 views

Poincaré duality for currents and non-closed forms

In page 8 of Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology by Szabo, the author claims that Poincaré duality holds for non-closed forms as long as the other form ...
2
votes
1answer
23 views

Fourier transform of $H(x-1)/x$

Consider $H(x-1)/x$ as a tempered distribution where $H$ is the Heaviside step function. I want to find an explicit form for its Fourier transform. Any ideas?
3
votes
1answer
62 views

Why are these distributions positive?

I am trying to understand some calculations in a paper by Sidney Coleman. He is showing that certain distributions are positive. The paper can be found here. What I am talking about is happening at ...
1
vote
1answer
18 views

Solve the following distributional differential equation: $(xT_f)' \equiv H$

As stated in the title, I want to solve the distributional differential equation $(\star)$ $$(xT_f)' \equiv H $$ $T_f \in (C_0^\infty)^*$ is a distribution induced by an arbitrary $f \in ...
0
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0answers
14 views

Show the following distributional equation: $v \delta'=v(0) \delta - v'(0) \delta$ for $v \in C^\infty$

As in the title stated I want to show that $$v \delta'=v(0) \delta - v'(0) \delta$$ in distributional sense where $v \in C^\infty$ and $\delta$ is the Dirac-Delta-Functional. We introduced it by ...
0
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0answers
16 views

Folland Exercise 9.20

In this problem, let $S'$ be the space of tempered distributions and $E' = \left\lbrace T \in D'(U): supp(T) \subset U, supp(T) compact \right\rbrace$. Suppose that $F \in S'$ and $G \in E'$. ...
3
votes
2answers
56 views

Fourier Transform Dirac Delta

I have recently learnt about tempered distributions, and how one can define the Fourier transform of a tempered distribution $v$ to be $\hat v$ so that $$\langle\hat v,\varphi\rangle=\langle v,\hat ...
0
votes
1answer
22 views

Show $supp(T_\mu) = supp(\mu)$ where $T_\mu(\phi) = \int_U \phi d\mu$ for all test functions $\phi$

So first I was able to show that $T_\mu$ is in fact a distribution. To show their supports are equal, I'll look at the complements, and so I need to show that the largest open set on which $T_\mu =0$ ...
1
vote
2answers
28 views

Fundamental Solution to 2nd Order ODE

I'm currently doing a problem with the fundamental solution for $$-u''+k^2u=f(x) \quad , \quad -\infty < x < \infty$$ I'm wondering if fundamental solutions are supposed to satisfy the ...
0
votes
1answer
22 views

Distributional derivative with discontinuities

Suppose that $f$ is continuously differentiable on $\mathbb{R}$ except at $x_1,\cdots,x_m$ where $f$ has jump discontinuities, and that its pointwise derivative $df/dx$ (defined except at ...
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0answers
34 views

Folland Exercise 9.21b

On $\mathbb{R}$, let $F$ be the constant function $1, G = \frac{d\delta}{dx}$, and $H = \chi_{(0,\infty)}$. Then $(F*G)*H$ and $F*(G*H)$ are well defined in $S'$ but are unequal. Without doing ...
0
votes
1answer
55 views

Decomposition of complex Radon measures

Suppose you have a complex Radon measure $\mu$, treated as a distribution. Then does every such Radon measure admit a decomposition of the form $\mu = \sum_{n=1}^\infty c_n \delta(x-\tau_n) + \hat f$ ...
0
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0answers
4 views

About test functions for supersolutions

Let $B_{1}$ the unit open ball in $\mathbb{R}^{n}$ and $u \in H^{1}(B_{1})$. For $k,m >0$, let $\bar{u} = u^{+} + k$ and $\bar{u}_{m} = \bar{u}$ if $u < m$ and $\bar{u}_{m} = k+m$ if $u \geq m$. ...
0
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1answer
54 views

Support of a Distribution

Let $U$ be a nonempty open subset of $\mathbb{R}^n$ and $\mu$ be a Radon measure on $U$. Define $$T_\mu(\phi) = \int_U \phi d\mu$$ for all $\phi \in D(U) = C_c^\infty(U)$. Prove that $T_\mu$ is a ...
2
votes
1answer
58 views

Space of test functions defined by norms

This is the problem assigned: So I know that a locally convex Hausdorff space is defined by a vector space and a family of seminorms. So is part $a$ just wanting me to show that $\|\phi\|_m$ is in ...
0
votes
1answer
10 views

Prove version of Bernsteins inequality $||\partial^{\alpha}f||_{L^{\infty}}\leq CR^{|\alpha|}||f||_{L^{\infty}}$

This is the question I am trying to answer, I am having difficulties understnading what is going on. My first question is there a typo in the hint, i.e should it be a new function $g=f\ast h_{1/R}$ ...
0
votes
1answer
30 views

Obtain a tempered distribution from $1/|x|$ by subtracting a multiple of $\phi(0)$

I am trying to show that for an appropriate choice of constants $c_{\delta}$ which diverge as $\delta \to 0$ a distribution $W\in \mathcal{S}'(\mathbb{R})$ can be defined by: $$ W(\phi)=\lim_{\delta ...
1
vote
0answers
19 views

Multiplying the PV$(\frac{1}{x})$ by $x$

I am trying to show that $x\text{PV}\left(\frac{1}{x}\right) = 1$ in the sense of distributions, that is $\langle x\text{PV}\left(\frac{1}{x}\right), \phi \rangle = \langle 1, \phi \rangle$ for all ...
10
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1answer
210 views

Proving that a family of functions converges to the Dirac delta.

For each $\epsilon > 0$, define $f_\epsilon:\mathbb R\to \mathbb R$ as follows: \begin{align} f_\epsilon(k) = \frac{1}{\pi}\frac{\epsilon}{\epsilon^2+k^2}. \end{align} How does one rigorously ...
0
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0answers
22 views

Distributional derivative of Indicator function $\times$ smooth function

I have a question about distributional derivative. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$. Suppose $\Omega_{1} \subset \Omega$ has the following property: $f \in ...
0
votes
1answer
14 views

Let $T : \mathcal{D}(\mathbb{R}) \to \mathbb{R}$ be given by $T(\phi) = |\phi(0)|$. Show that $T$ is not a distribution.

As the title states, I wish to show that $T(\phi) = |\phi(0)|$ is not a distribution. I assume I need to show that the bound $|T(\phi)| \leq C \sum_{|\alpha| \leq n} ||D^{\alpha}\phi||_{L^{\infty}}$ ...
2
votes
1answer
45 views

Second derivative of the delta function

Is the second derivative of the delta 'function' even? My intuition tells me yes, and my calculation relies on delta''(-x) = delta''(x).
1
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0answers
43 views

Is there a Plancherel-type identity for generalized Fourier Transforms?

Let $S$ be in $\mathcal{T}$, the set of tempered distributions, and $\mathcal{F}S$ be its Fourier Transform. Is there some relationship for such distributions, analogous to the Plancherel Theorem for ...
4
votes
2answers
722 views

Convergence of test-functions is not induced by any metric.

By $\mathcal{D}(\mathbb{R})$ we denote linear space of smooth compactly supported functions. We say that $\{\varphi_n:n\in\mathbb{N}\}\subset\mathcal{D}(\mathbb{R})$ converges to ...
1
vote
1answer
37 views

Second order differential equation with Heaviside function

I have a differential equation of the form $$y''(x) - a y(x) + b \theta(c - x) = 0, \quad y(0) = 0, \quad \lim_{x \to \infty} y(x) = 0,$$ where $a$, $b$, $c$ are some constants and $\theta(с - x)$ is ...
1
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0answers
47 views

Definition of the convolution with tempered distributions and Schwartz function

If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=\langle \tau_x \widetilde{\varphi} , u \rangle$, where we place ...
0
votes
1answer
30 views

Let $\phi\in\mathscr{D}$. Then $f\phi\in\mathscr{D}$ for every smooth function $f$.

Let $\phi\in\mathscr{D}$, where $\phi$ is a test function and $\mathscr{D}$ is the set of all test functions. Then $f\phi\in\mathscr{D}$ for every smooth function $f$. This one seems...trivial. So ...
0
votes
1answer
23 views

Properties of convergence on the set of test functions

I'm trying to prove the properties of convergence on the set of test functions, $\mathscr{D}$, but the following is giving me some problems. Let $\phi_n\to\phi$ and $\psi_n\to\psi$ on $\mathscr{D}$. ...
1
vote
1answer
18 views

Prove that the functional $F$ on $\mathscr{D}$ defined by $\langle F, \phi\rangle=\int_{\mathbb{R}^n} f\phi$ is a distribution.

Let $f$ be a locally integrable function on $\mathbb{R}^n$. Prove that the functional $F$ on $\mathscr{D}$ defined by $\langle F, \phi \rangle = \int_{\mathbb{R}^n} f\phi$ is a distribution, where ...
0
votes
0answers
28 views

Show that $f_n\to 0$ in the distributional sense.

Let $f_n(x)=\sin{(nx)}$. Show that $f_n\to 0$ in the distributional sense. I know that this is true only if $\langle f_n,\phi\rangle=\int_{\mathbb{R}^n} f_n\phi\to \int_{\mathbb{R}^n} f\phi=\langle ...
0
votes
0answers
49 views

Integrating with a Dirac delta function $\delta(x-a)$ when $a$s not in the domain of integration?

The delta function has the fundamental property that \begin{align} \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) \end{align} and, in fact, \begin{align} ...
0
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0answers
34 views

About Semiclassical Analysis and other

I read something about this theory. I honestly do not care to find out the link between quantum mechanics and general relativity, because it's too much for me. But I have seen that there are still ...
1
vote
1answer
80 views

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$?

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$? Where $\mathcal{D}(\Omega)$ is the space of test functions with support compact and $\mathcal{D}'(\Omega)$ is the ...
0
votes
0answers
23 views

Representation of the delta distribution as an element of the dual of $H^1$

I'm working with some Sobolev spaces and I just wanted to consider the elements of $H^{-1}$ as elements on $H^1$ (Riez Theorem). Since the delta function $\delta(f) = f(0)$ is an element of the dual ...
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0answers
22 views

Show that the following are equivalent for $u \in C^{1}(0,1)$

Let $u \in C^{1}(0,1)$. Prove that the following are equivalent: (a) $u \in W^{1,1}(0,1)$ (b) $u' \in L^{1}(0,1)$ (where $u'$ means the derivative in the usual sense) (c) $u \in BV(0,1)$ My try: ...
0
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0answers
14 views

Transport PDEs with mixed linear and nonlinear terms and distribution solutions

This question is concerned with the theory of solutions of first order transport-type PDEs that are linear in some variable and nonlinear in others. E.g. this beauty: $\frac{\partial u}{\partial ...
1
vote
2answers
233 views

cutoff function vs mollifiers

Q1. What are cutoff function? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions? And how they differ from mollifiers I did check ...
3
votes
2answers
61 views

Show that $f(\phi)=\sum_{k=0}^\infty \phi^{(k)}(k)$ for $\phi \in D$ has no finite order

Let $\phi \in D:=D(K)=C^\infty_0(K) $ be a test function and let $f \in D^*=\{f:D \to \mathbb{R} : f \text{ bounded and linear} \} $ be a distribution. A distribution has finite order if: $$\exists ...
1
vote
1answer
31 views

Singular integral operator

i got the following problem to solve. Let $0 < \alpha < 1$, $L \in L_\infty([0,1]^2)$, $D = \{(x,y) \in \mathbb{R}^2: x = y\}$ the diaagonal of $\mathbb{R}^2$ and $k:[0,1]^2 \setminus D \to ...
1
vote
1answer
47 views

Rellich's theorem fails at the end point.

If $\{f_k\}\subseteq H^s(\Bbb{R}^d)$ is a bounded sequence and support of each $f_k$ is contained in a common compact set, then there exists a subsequence that converges in $L^q$ for $\forall$ $q, ...
0
votes
0answers
20 views

Bounded sequence in Sobolev space has a convergent subsequence

Suppose $\{\phi_k\}$ are a bounded sequence of distributions on in $H^s(\Bbb{R^d})$, where $0<s<\frac{d}{2}$ and $H^s$ is the Sobolev space. Then the sequence has a subsequence that converges in ...
0
votes
1answer
17 views

Definition of homogeneous distribution.

I ran into the following definition: If $u$ is a distribution on $\mathbb{R}^d$, then $u$ is called homogeneous of order $m$ if $u(\lambda x) = \lambda^m u(x)$, $x\in\mathbb{R}^d$. But $u$ is not ...
3
votes
0answers
212 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
2
votes
1answer
877 views

function a.e. differentiable and it's weak derivative

Note - I am just starting to learn about theory of distributions, so this may be a trivial question, if so I'd be grateful for a reference, nevertheless the question is the following: suppose I have a ...
8
votes
1answer
664 views

Sequential and topological duals of test function spaces

Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
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0answers
28 views

Proof that inequality holds

Theorem: Let $u \in D'(\Omega)$ and $K \subset \Omega$, $K$ compact $$\exists \lambda \in \mathbb{N} \text{ and } c \geq 0 \text{ such that } \\ |\langle u, \phi \rangle| \leq c \sum_{|a| \leq ...