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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16 views

How to define a variable which is an integral involving cauchy principal value inside?

How to define a variable which is an integral involving cauchy principal value inside in any computer programming language? I want to know how to break down the procedure step by step from a ...
2
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2answers
63 views

A doubt about the vectorial topology on $\mathcal{D}(\Omega)$

We denote with $\mathcal{U}_0$ the family of all subsets $U \in \mathcal{D}(\Omega)$ convex and balanced such that $U \cap \mathcal{D}_K(\Omega) \in \mathcal{T}_K$, where $\mathcal{T}_K$ is the ...
3
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1answer
88 views

How to show for a distribution $T$ and a test function $\varphi,~~T'[\varphi]\equiv -T[\varphi']\;?$

For a generalized function $T,$ we define $$T'[\varphi] ~≡~ −T[φ']~~~~~~\forall φ ∈ \mathcal D(Ω).$$ where $\mathcal D(\Omega)$ denotes the test function space. I'm not getting how they ...
4
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1answer
237 views

Definition of the convolution with tempered distributions and Schwartz function

In the book where I'm studying there is the following exercise. 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)=...
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2answers
23 views

Representation of functional on overlapping areas

I have given a functional $l$ on $C_c^\infty(\mathbb{R}^n)$. Now let's assume that for any $p \in \mathbb{R}^n$ we have a neighborhood $V_p$ and a $2\pi$-periodic $C^\infty$-function $u_p$ on $\mathbb{...
1
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1answer
35 views

Derivative of principal value distribution $1/x^2$ is equal to finite part distribution $-1/x^2$?

Finite part (Partie finie) of the mapping $x \mapsto \frac{1}{{{x^2}}}$ is a regular distribution defined by $$\left\langle {{\text{Pf}}\frac{1}{{{x^2}}},\varphi } \right\rangle = \mathop {\lim }\...
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1answer
20 views

Distributional derivatives for functions that is continuous but nowhere differentiable

It is well known that the Brownian motion is an example of functions that is continuous but nowhere differentiable. In addition, its distributional derivative can be interpreted in the way mentioned ...
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3answers
90 views

What is the theory of distribution that makes possible to calculate Fourier transform of the Sine function?

I am an Engineering student. Sine wave function is a power signal present from $-\infty$ to $+\infty$. I have have read that because of distribution theory,the Fourier transform of the Sine function ...
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0answers
29 views

Is it true that $|∇u(x)|^2\chi_\Omega=|\nabla (u \chi_\Omega)|^2$

Let $u\in L^\infty(\Omega)\cap H^1(\Omega)$ with $\Omega$ open, bounded and regular (as you wish) domain of $\mathbb{R}^N$. Is it true that $$ \int_\Omega |\nabla u(x)|^2 \mathrm{d} x= \int_{\mathbb{...
5
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1answer
60 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
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2answers
81 views

Identity for rescaled Dirac Delta, $\delta(kx)$

I´m trying to proof the following Statement. $$\delta(kx)=\frac{1}{|k|} \delta(x).$$ I already tried to proof and I got this. $$u=kx \Rightarrow x=\frac{u}{k},dx=\frac{1}{k} du \\ \int_{-\infty}^\...
1
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1answer
25 views

Weak solutions of Navier-Stokes are square integrable distributions?

I'm reading Lemarie-Rieusset's book Recent developments in the Navier-Stokes problem and have the following issue: he defines a weak solution to the Navier-Stokes equations on $(0,T)\times\mathbb R^d$ ...
0
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0answers
22 views

How do we show the inequality for $p=\infty$?

How can we show the inequality for $p=\infty$ ? Since $\overline{u} \in W^{1, \infty}(\mathbb{R}^n)$ we have that $\overline{u}'$ exists and $\overline{u}, \overline{u}'$ are essentially bounded. ...
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0answers
89 views

Anti-derivative of Dirac distribution over $\mathbb{R}^{n}$.

I want to determine an $f\in C(\mathbb{R}^{n})$ and a multi-index $\alpha$ such that $\partial^{\alpha}f=\delta$. I found an example in some lecture notes which claimed that for $$f(x)=\frac{1}{\...
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1answer
91 views

Construct extension of function

If $u \in W^{3,p}(\mathbb{R}^{+})$ how can we construct the catoptric extension $\overline{u}$ of $u$ in $\mathbb{R}$ (reflection) such that $\overline{u} \in W^{3,p}(\mathbb{R})$ ? EDIT: By setting $...
2
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1answer
80 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
18 views

Show property of convolution

Proposition: Let $ u, v \in L^2(\mathbb{R}^n) $ then $ \widehat{u \ast v}=\widehat{u} \cdot \widehat{v}$. Proof: We want to show that $\mathcal{F}^{-1} (\widehat{u} \cdot \widehat{v})=u \ast v $. We ...
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0answers
45 views

Show $L^2$-convergence

Lemma: Let $\psi \in C_C^{\infty}(\mathbb{R}^n), \psi \geq 0, \int \psi(x)dx=1, \psi_{\epsilon}(x)= \epsilon^{-n} \psi{\left( \frac{x}{\epsilon}\right)}, \epsilon>0$. Let $f \in L^2(\mathbb{R}^n)$ ...
3
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3answers
47 views

How to solve a differential equation with a distributional free term?

I tried to solve this type of differential equation $$y'' + y = \delta + \delta' .$$ I tried using the Laplace Transform, but I'm stuck at that $\delta$ (Dirac function). The only thing I know is ...
3
votes
1answer
1k views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, k\...
3
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1answer
66 views

Find $\sum_{k \geq 1} e^{itk}$ in the sense of distribution - $\delta(x-a)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i(x-a)t}dt$

I have to solve $Z(t)=\sum_{k \geq 1} e^{itk}$ in the sense of distribution (generalized function), i.e., $<\sum_{k \geq 1} e^{itk}, \varphi>$, where $\varphi$ is a test function. So far, by the ...
2
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1answer
45 views

Integral of delta function and the constant for fund. solution to laplace's eq

When finding the fundamental solution to Laplace's eqn, i.e. $G$ such that $\Delta G = \delta$ a constant has so be solved for. How I have seen this done is by finding $c$ so that $\int_{D(0,\epsilon)...
2
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1answer
75 views

Integral of Dirac delta function from zero to infinity

I know that: $$\int_{-\infty}^{+\infty} \mathrm{d}t \, f(t) \delta(t) = f(0)$$ However, I cannot figure out the result of the integral below: $$\int_{0}^{+\infty} \mathrm{d}t \, f(t) \delta(t) = ?$$ ...
3
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1answer
53 views

Proof that estimate holds

Theorem: Let $U$ be a bounded , open subset of $\mathbb{R}^n$ , and suppose $\partial{U}$ is $C^1$. Assume $1 \leq p<n$, and $u \in W^{1,p}(U)$. Then $u \in L^{p^{\ast}}(U)$ , with the estimate $||...
2
votes
1answer
95 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: $||\widehat{u}||_{L^2(\mathbb{R}...
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1answer
47 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 ...
2
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2answers
50 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, \beta\in\mathbb{...
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0answers
19 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 Dirac-$\delta$-...
2
votes
1answer
82 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
20 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 ...
1
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1answer
59 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) ...
1
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1answer
42 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 -x)\...
2
votes
1answer
29 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
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 ...
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2answers
66 views

Proof of uniqueness about distribution in Rudin's

I'm reading Functional Analysis by Rudin, and have trouble understanding a part of the proof of theorem 6.33, in page 174. This theorem states an one-to-one relationship between a linear continuous ...
0
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1answer
18 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 $W^{1,...
2
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1answer
75 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
57 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 $S(\mathbb{R}^n)$ $S(\mathbb{R}^...
0
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1answer
38 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 ...
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2answers
54 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} d{\...
0
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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} \delta_n(x)\phi(x)\mathrm{d}...
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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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2answers
99 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.
6
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1answer
76 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
24 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
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1answer
63 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
20 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 L_{\text{...
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0answers
18 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
22 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 $G \in E'$. ...