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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1answer
39 views

Derivatives of the Dirac delta function

From what I understand the Dirac's Delta derivatives have the meaning $$\int_{-\infty}^{\infty}\delta^{(k)}(x)\phi(x)dx=(-1)^k\int_{-\infty}^{\infty}\delta(x)\phi^{(k)}(x)dx$$ Assuming, of course that ...
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1answer
47 views

$H_m(\mathbb{R}^n)$ , the completion of $C_C^{\infty}(\mathbb{R}^n)$

Theorem: Let $m$ be a positive integer. Then $H_m(\mathbb{R}^n)=\{ u \in D'(\mathbb{R}^n): D^{\alpha} u \in L^2(\mathbb{R}^n), |\alpha| \leq m\}$ $\to ||u||_{H_m}^2=(2 \pi)^{-n} \int (1+|\xi|^2)^m ...
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0answers
16 views

Distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi '>$ in term of Dirac-$\delta$-function

Personal question : Could it possible for the distribution distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi'>$ to be expressed in term of the ...
3
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1answer
76 views

$\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ - Theory of distribution

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
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0answers
16 views

Relation of rate of decay of a function with width of peaks of its Fourier transform

Consider a function $f(t)=\theta(t)e^{-\sigma_0 t}\sin(\omega_0 t)$, where $\theta(t)$ is $1$ for positive $t$ and $0$ for negative $t$. Its Fourier transform can be easily computed. It has the ...
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2answers
42 views

A characterization of tempered distributions

The Schwartz space on $\mathbb{R}^n$ is the function space $$ S \left(\mathbf{R}^n\right) = \left \{ f \in C^\infty(\mathbf{R}^n) : \|f\|_{\alpha,\beta} < \infty,\, \forall \alpha, ...
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0answers
82 views

$\langle w,\varphi\rangle =\int_{\mathbb{S}^1} \left(\sum_{k \geq 1} e^{itk}\right) \varphi(t) \, dt$ - Generalized function

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
2
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1answer
65 views

Parseval's identity holds

Theorem: If $u \in L^2(\mathbb{R}^n)$ then the Fourier transform $\widehat{u} \in S'(\mathbb{R}^n)$ is a $L^2(\mathbb{R}^n)$ function and the Parseval's identity holds: ...
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1answer
15 views

$<f_a, \varphi> = \int_{-\infty}^{\infty} \frac{\varphi(x)}{|x|} dx$ know that $\varphi(0)=0$ - Generalized function

Question : Show that $<f_a, \varphi> = \int_{-\infty}^{\infty} \frac{\varphi(x)}{|x|} dx$ for any $\varphi \in D(\mathbb{R})$ for which $\varphi(0)=0$. I am a little bit confused how to solve ...
2
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1answer
28 views

$\lim_{n \to \infty} \langle f_n, \varphi \rangle$ - Generalized function

Question : Let $f_n$ be the distribution $<f_n,\varphi>=n(\varphi(\frac{1}{n})-\varphi(\frac{-1}{n}))$. What distribution is $\lim_{n \to \infty} <f_n, \varphi>$ ? First try : $\lim_{n ...
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1answer
34 views

Can you recover a distribution from mollification?

Let $f\in \mathcal S'(\mathbb R)$ be a Schwartz distribution. Given $\rho \in C^\infty_c(\mathbb R)$ define the convolution as the function $$x\mapsto (f\ast\rho)(x):=\langle f, \rho (\cdot ...
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1answer
15 views

can I get weak convergence in sobolev spaces from convergence of distributions

my question is the following. Given a sequence $(f_k)_k$ in $W^{1,q}(\Omega)$ with $q \in (1,\infty)$ and $\Omega \subseteq \mathbb{R}^n$ open and bounded. If I want to show $f_k$ converges in ...
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2answers
48 views

Relation between Dirac function and inverse fourier transform of 1

According to my notes, it holds that $\delta=(2 \pi)^{-n} \widehat{1}$. How do we get the equality? We have that $\delta=\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \widehat{\delta}(\xi) e^{i x \xi} ...
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1answer
37 views

calculate Fourier Transformate

i have the following exercice: Let for all $x \in \mathbb{R},$ $f(x)= \cos x$ and $g(x)= \sin x$. Calculate $T=f \delta' + g \delta''$ for this question, i find $T=3 \delta$. Calculate the Fourier ...
2
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1answer
73 views

Show that $\int h_n'(x) \varphi(x)\, dx \to \langle \delta, \varphi\rangle$ - Generalized functions theory

In the book Partial Differential Equations by Robert Strichartz, there's an exercise (#$1$, page $9$) that I am not quite sure how to solve. Is there anyone could give me the principal steps how to ...
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2answers
49 views

Show inclusion and embedding

I am looking at the following theorem: $C_C^{\infty}(\mathbb{R}^n) \subset S(\mathbb{R}^n)$ and the embedding is continuous. $C_C^{\infty}(\mathbb{R}^n)$ is dense in $S(\mathbb{R}^n)$ ...
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1answer
52 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) ...
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1answer
49 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} ...
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2answers
88 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.
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1answer
35 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!
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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?
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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 ...
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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 ...
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0answers
17 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
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2answers
59 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 ...
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1answer
25 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 $T_\mu =0$ ...
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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 ...
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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
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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2answers
29 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 ...
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1answer
57 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 ...
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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 ...
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1answer
61 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 ...
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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
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1answer
57 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 $\mu = \sum_{n=1}^\infty c_n \delta(x-\tau_n) + \hat f$ ...
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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 ...
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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 ...
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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 ...
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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 $|T(\phi)| \leq C \sum_{|\alpha| \leq n} ||D^{\alpha}\phi||_{L^{\infty}}$ ...
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1answer
46 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).
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44 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 ...
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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 ...
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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 ...
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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 $\mathscr{D}$. ...
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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 ...
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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 ...
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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} ...
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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 ...
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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: ...