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Questions tagged [distribution-theory]

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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Norm on the space of rapidly decreasing and continuous functions

In P.Malliavin’s book "Integration and probability" a continuous function $f$ defined on $\mathbb{R}^n$ is said to be of rapidly decrease if for all integer $m$ we have that the mapping $(1+\...
G2MWF's user avatar
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1 answer
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Convolution of a distribution and a function with tensor product

I am stuck in writing the convolution $\star: D' \times D \to E$ defined as $$(f \star \phi)(x) := \langle f_y, \phi(x-y) \rangle$$ as a tensor product. Could someone help me out? I was told with the ...
AlexAsks's user avatar
4 votes
1 answer
65 views

Is it true that $\widehat{(\delta_{x_{0}}\otimes T)} = \hat{\delta}_{x_{0}}\otimes \hat{T}$?

For a fixed $x_{0} \in \mathbb{R}$ consider the Dirac delta distribution $\delta_{x_{0}}$. Its Fourier transform is given by $\hat{\delta}_{x_{0}}(p) = e^{-px_{0}}$, in the sense that $\hat{\delta}_{...
InMathweTrust's user avatar
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0 answers
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Integral representation of Dirac delta? [closed]

Is there any integral representation of the Dirac delta distribution as follows: $$\delta(x-x_{0}) = \lim_{n\to \infty}\int dx f_{n}(x-x_{0})\frac{1}{\sqrt{|x|^{2}+m^{2}}}$$ for a fixed parameter $m &...
Idontgetit's user avatar
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1 vote
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37 views

Proof of Paley-Wiener Theorem

I'm trying to understand the proof of the following version of Paley-Wiener theorem under the additional assumption $f \in L^2$: I understood the part $(2) \Rightarrow (1)$ but I couldn't follow a ...
heyy's user avatar
  • 119
2 votes
0 answers
51 views

Understanding spaces of negative regularity

Let $C^k(\mathbb{R}^n$) be the space of functions with $k$ continuous derivatives, and $H^s(\mathbb{R}^n)$ the Sobolev space $W^{2,s}$. Their dual spaces are commonly denoted as $C^{-k}$ or $H^{-s}$. ...
CBBAM's user avatar
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2 votes
0 answers
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Solving differential equations with sharp corners in the form of Dirac Delta function

Suppose we have the following well behaved differential equations of time $t$ $$ \ddot y + (-\frac{\dot w}{w}+2w)\dot y + w^2y = w^2x.$$ By well behaved it is meant that change in $w$ and scaling the ...
Magemathician's user avatar
1 vote
0 answers
42 views

Characterization of tempered distributions on the sphere $S^n$

Let $\mathbb{R}^n$ be $n$-dimensional Euclidean space and $S^n$ be the $n$-sphere. If we fix a point $x \in S$, the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ can be characterized as the subspace of $...
Keith's user avatar
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-5 votes
1 answer
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A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation?

A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation? Posted later after comments: In summary, I am trying to understand what ...
Joako's user avatar
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Question about Rudin's Functional Analysis

I have a question about the proof of theorem 8.5 in Rudin's Functional Analysis. In the equation between (10) and (11), it gives $(P(D)E)(\phi)=E(P(-D)\phi)=u((P(-D)\phi)^{\vee})=u(P(D)\phi^{\vee})=\...
xyz's user avatar
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1 vote
1 answer
68 views

Failure of Fubini when integrating in the sense of distributions

Similar questions have been asked, but I find it hard to apply it to this case. The following is a physics motivated problem and should illustrate how Fubini fails. The main question is then: How do I ...
Confuse-ray30's user avatar
1 vote
0 answers
28 views

Limit behaviour of a distribution

Consider the distribution, with $a>0,p>0$ $$ F(x)=\left\{\begin{matrix}\frac{J_2(a\sqrt{p^2-x^2})}{x- p} & -p<x<p \\ 0 & \text{otherwise}\end{matrix}\right. $$ I claim that $F$ ...
Wouter's user avatar
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1 vote
1 answer
81 views

Is $e^x \cos(e^x) f(x)$ absolutely integrable on $\mathbb{R}$ for any Schwartz function $f$?

In this ME post, it is said that the function $e^x \cos(e^x)$ is a tempered distribution on $\mathbb{R}$. Namely, we have \begin{equation} \int_{\mathbb{R}} e^x \cos(e^x) f(x) dx = -\int_{\mathbb{R}} \...
Keith's user avatar
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1 vote
0 answers
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Passing from $\mathcal{D}$ to $\mathcal{S}$ using a density argument and extra condition

I saw a proof for the following statement for the space of Schwartz functions $\mathcal{S}$: $$ \varphi \in \mathcal{S}: \int \varphi = 0 \iff \exists \Phi\in \mathcal{S}: \Phi' = \varphi, $$ which ...
Taleofwoe's user avatar
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Possible to define an inner product on tempered distributions of compact support?

I am trying to understand why, in the context of reproducing kernel Hilbert spaces, there seems to always be a square-energy restriction on bandlimited functions in the Paley-Wiener space. (I get why ...
iamnotacrackpot's user avatar
1 vote
0 answers
40 views

Density of Schwartz distributions in the space of distribution

Let $S(R^3)$ and $D(R^3 )$ be the space of Schwartz function and test function respectively, $S'(R^3)$ and $D'(R^3)$ be their duals. I want to understand what "$S'(R^3)\subset D'(R^3)$ and the ...
Alucard-o Ming's user avatar
1 vote
1 answer
39 views

Finding a Closed Form Expression for a Distribution Defined by an Integral Involving Sine and Bessel Functions

I am seeking a closed form expression for the following distribution: $$ D(t,x) = \int_0^\infty d\omega\, \omega^2 \sin(\omega t) J_0(\omega x), $$ where $J_0(x)$ is the Bessel function of the first ...
Adam Teixidó Bonfill's user avatar
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0 answers
45 views

Show $\text{p.v.}\frac{x_{i}}{|x|^{n+1}}$ is tempered distribution on $\mathbb{R}^n$

Suppose dimension of $\mathbb{R}^n$ is not smaller than $2$ and $\varphi\in\mathcal{S}(\mathbb{R}^n)$, for $1\leq i\leq n$, define the functional $$\langle\text{p.v.}\frac{x_{i}}{|x|^{n+1}},\varphi\...
Cunyi Nan's user avatar
  • 742
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29 views

Weak* convergence in Hilbert Spaces

I have two Hilbert spaces $H,V$ s.t. $H \subset V$ (You could think, for example, at $H^1$ and $L^2$). Suppose i have two subsequences, $u_m(t)$ and $v_m$, s.t. $u_m(t) \to u(t)$ weakly* in $L^\infty(...
Lucio Rosi's user avatar
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0 answers
44 views

Is $u(x)=\frac{1}{|x|^{\alpha}}$ in $W^{1,p}(B_1(0))$?

Consider a function $$ u(x)=\frac{1}{|x|^{\alpha}} \quad x\in B_1(0) \subset \mathbb{R}^N. $$ I should find condition about $p, N, \alpha$ for $u$ to be in $W^{1,p}(B_1(0))$. Following different books ...
Shiva's user avatar
  • 133
1 vote
0 answers
25 views

Weak derivative operator is not sequentially continuous in $L^1_{\text{loc}}(\Omega)$

Suppose we have a sequence $(u_n)_{n\in\mathbb{N}}$ in $L^1_{\text{loc}}(\Omega)$ and $u\in L^1_{\text{loc}}(\Omega)$ such that $u_n\to u$ in $L^1_{\text{loc}}(\Omega)$ and all the functions $u_n$ and ...
Nepal Draus's user avatar
0 votes
1 answer
41 views

Problem proving tempered distibution

Given $$ T = (x^2+x+1)\delta' $$ How can I prove that $T$ is a tempered distribution? The step that I took are as follows. I define $ f(x)=x^2+x+1 $ and I used Leibniz's rule that states: $$ \dfrac{d}{...
Hamza Amine's user avatar
1 vote
0 answers
74 views

Does a generalization of the Sokhotski-Plemelj Formula to four (or higher) dimensions exist?

The Sokhotski-Plemelj Formula states \begin{equation} \frac{1}{x \pm \textbf i \eta} = \mathcal{P} \left(\frac1x\right) \mp \textbf i\pi \delta(x) \end{equation} where this expression has to be ...
Steven's user avatar
  • 11
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0 answers
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Show that for all $\phi \in V , \int_a^b u(x)\phi (x)=0$

Let $V=\{ \phi \in C^{\infty}_c(]a,b[) : \int_a^b \phi (x)=0\}$ and $u\in L^1(]a,b[)$ such that for all $\phi \in C^{\infty}_c(]a,b[)$ we have $$\int_a^b u(x)\phi' (x)=0$$ Show that for all $\phi \...
gebafe's user avatar
  • 21
2 votes
2 answers
57 views

Fourier transform of $f(x)=x\mathcal{X}_{[a,b]}$, for $a<b$ reals.

As stated, this problem is pretty straigth forward. Using the normal definition of the Fourier transform I get, $$\widehat{x\mathcal{X}_{[a,b]}}=\int_{a}^{b}xe^{-2\pi i x \xi}dx=\frac{e^{-2\pi i a \xi}...
Tutusaus's user avatar
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1 vote
0 answers
51 views

Convergence of moments implies convergence of the measure in the space of tempered distributions?

Let $\mathcal{S}(\mathbb{R}^d)$ be the Schwartz space and $\mathcal{S}'(\mathbb{R}^d)$ be the space of tempered distributions. Denote by $\{\mu_k\}_{k=1}^\infty$ a sequence of Borel probability ...
Keith's user avatar
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1 vote
1 answer
76 views

Proof review Tempered Distributions: $(T*\varphi)^\wedge = (2\pi)^{n/2} \hat{\varphi}\hat{T}$ and computing $(x^\alpha)^\wedge$

I am learning about tempered distributions on schwartz space. But there are 2 questions in this text where Im unsatisfied with my solution... I ask kindly for your review of my solutions below, and if ...
NazimJ's user avatar
  • 3,266
0 votes
1 answer
47 views

Contour integral of complex generalized function

I'm reading a derivation of the Green function for the hyperbolic PDE and got to the point where I need to evaluate the integral: $$\int_{-\infty}^{+\infty}e^{i\omega t}(\frac{1}{\omega-kc-i0}-\frac{1}...
Krum Kutsarov's user avatar
0 votes
1 answer
34 views

$L^p(\mathbb{T})$ is identified with tempered distibution (periodic)?

i know that, the space $L^p(\mathbb{R}^n)$ canbe idenfitied with a tempered distributuion. Can be the space $L^p(\mathbb{T})$ identified with a tempered distribution (periodic?)?
eraldcoil's user avatar
  • 3,660
1 vote
0 answers
24 views

CLT for exchangeable and symmetric dependent Bernoulli variables

Consider a finite set of $n$ dependent Bernoulli random variables $X_1$, $X_2$, ..., $X_n$ that are symmetric and therefore exchangeable. This means: Each variable $X_i$ has the same probability of ...
Emma's user avatar
  • 11
0 votes
0 answers
41 views

My proof of $\delta_0\left(t^2-|x|^2\right)=\frac{1}{2 \sqrt{2} t} \mathrm{~d} \sigma_{C_0^{+}}(t, x)$

Here is the proof of Lemma 4.10. On $\mathbb{R}^{1+3} \backslash\{0\}$, we have the identity $$ \delta_0\left(t^2-|x|^2\right)=\frac{1}{2 \sqrt{2} t} \mathrm{~d} \sigma_{C_0^{+}}(t, x) $$ where $C_0^{...
YuerCauchy's user avatar
3 votes
2 answers
74 views

Differentiating Dirac delta with product rule

I have here an equation. $$ h'(t_2) \delta(t_1 - t_2) = [h(t_2) - h(t_1)] \delta'(t_1 - t_2) $$ I checked the equality by integrating both sides with a test function. $$ \int d t_1 \phi(t_1) \ldots \...
Bio's user avatar
  • 1,108
0 votes
0 answers
30 views

The $L^\infty$ boundedness of the resolvent of Harmonic Oscillator in terms of seminorms

I am reading Watanabe's book "Lectures on Stochastic Differential Equations and Malliavin Calculus". In Page 48, it is said that by means of (damped) harmonic oscillator $1+|x|^2-\Delta$, we ...
ze min jiang's user avatar
1 vote
0 answers
40 views

Distribution induced by a Radon measure

Let $\Omega \subset \mathbb{R}^N$ be open, and consider a distribution $T\in \mathcal{D}’(\Omega)$. I should prove the following statement. The distribution $T$ is a linear combination of Radon ...
Shiva's user avatar
  • 133
1 vote
0 answers
45 views

DiracDelta/x = -DiracDelta'? - Use and Correctness of Statement [duplicate]

One property of the Dirac Delta Distribution is $x \delta'(x) = -\delta(x)$ because of $\int x \delta'(x) f(x) dx = -\int \delta(x) (xf(x))' dx = -\int \delta(x) (f(x)+xf'(x)) dx = -\int \delta(x) f(x)...
theta_phi's user avatar
  • 147
0 votes
0 answers
36 views

Derivation of sifting property of Dirac Delta Distribution from Dirac sequence, one property isn't used

I want to start from the properties of the dirac sequence to derive its defining characteristic. To show: $\int_{-\infty}^\infty \delta(x) f(x) dx =f(0)$ Dirac sequence $\delta(x)_{n \in \mathbb{N}}$: ...
theta_phi's user avatar
  • 147
1 vote
1 answer
31 views

Laplace Transform of Derivative by Limit and Dirac Delta Distribution

The Laplace Transformation of the Derivative is given by $\mathcal{L}[f'(x)](s) = \int_0^\infty f'(x) e^{-s x} dx = s \mathcal{L}[f(x)](s) - f(0)$, which can be easily shown by partial integration. I ...
theta_phi's user avatar
  • 147
0 votes
1 answer
35 views

Why the test function space is countable union of $\mathcal{D}_{K_i}$

In the Rudin's functional analysis, in remark 6.9, It says there is countable collection of sets $K_i \subset \Omega$ such that $\mathcal{D}(\Omega)=\cup \mathcal{D}_{K_i}$ $\mathcal{D}(\Omega)$is the ...
xyz's user avatar
  • 709
1 vote
0 answers
23 views

Tempered distribution from locally integrable function

My professor said that if a function is $f \in L^1_{\text{loc}}\left(\mathbb{R}\right) $ . Then its associated distribution : $T_f$ is not a tempered distribution , we defined a tempered distribution ...
Knowledge Seeker's user avatar
0 votes
0 answers
50 views

What is the meaning of this definition?

Let $\mathscr{S}(\mathbb{R}^{d})$ the Schwartz space of rapidly decreasing functions and $\mathscr{S}'(\mathbb{R}^{d})$ the space of tempered distributions. Let $f$ be such that $f^{\gamma} \in C^{\...
InMathweTrust's user avatar
0 votes
0 answers
43 views

Does the Heaviside function belong to $W^{s,p}$ for some $s>0$?

Let us consider $n$-fold tensor product of Heaviside functions: \begin{equation} H(x_1, \cdots, x_n) := \prod_{i=1}^n \chi_{[0,\infty)}(x_i) \end{equation} Then, for any bounded open set $\Omega \...
Keith's user avatar
  • 7,829
0 votes
0 answers
31 views

Understanding different norms in the p-Wasserstein distance

The generalized p-Wasserstein distance, for $p\geq 1$, is given by $$d_W(Q_1,Q_2):=inf \left\{\int_{\Xi_2}||\xi_1-\xi_2||^p \Pi(d\xi_1,d\xi_2)\right\}$$ where $\Pi$ is the joint distribution of $\xi_1$...
osi41's user avatar
  • 99
1 vote
0 answers
33 views

How to prove that any Schwarz distribution is can be represented by hyper-function?

As far as I understand hyper-functions are generalizations for distributions (at least in 1-dimensional case and for some "good enough" definition of distribution...). What is the simplest ...
uhbif19's user avatar
  • 360
2 votes
1 answer
75 views

Bound of a compactly supported distribution

$u: \mathrm{C}^{\infty}(\Bbb R) \rightarrow \mathbb{R}$ is a linear map $$u(\varphi):=\sum_{j=1}^{\infty} \frac{\varphi\left(\frac{1}{j}\right)-\varphi\left(-\frac{1}{j}\right)}{j}$$ The question is ...
hbghlyj's user avatar
  • 3,047
1 vote
1 answer
66 views

Positive integral appears negative after applying the convolution theorem

I am Fourier transforming momentum representation of quantum mechanical wave functions to the position representation. In the weak sense (tempered distribution) we have $$\int_{\mathbb{R}^3} d^3k \ e^{...
Rasmus's user avatar
  • 514
4 votes
1 answer
121 views

How are polynomials of tempered distributions defined?

I am reading a textbook on rigorous quantum mechanics and quantum field theory in which there often appears statements such as Let $A(\phi)$ be a polynomial function defined on $\mathcal{S}'(\mathbb{...
CBBAM's user avatar
  • 6,295
1 vote
0 answers
36 views

A proof of a form of the Poincare Lemma:

If $\omega$ is a closed $k-$current on the interior of $S_n:=\{x \in \Bbb{R}^n: 0 \leq x_i \leq 1, 0 \leq x_1+...+x_n \leq 1\}.$ Then $\omega = d \eta$ where $\eta$ is an extendible $(k-1)-$current or ...
homosapien's user avatar
  • 4,213
1 vote
0 answers
100 views

Why is $u\in H^1$ enough s.t. $\int \nabla u\cdot\nabla v$ is a well defined expression?

Definitions Consider a distribution $T\in \mathcal{D}'(\Omega)$, then we define the distributional derivative by "duality" (Adjoint operator etc.), meaning: $$T'(\phi):=-T(\phi')\ \ \forall\...
user1313292's user avatar
0 votes
0 answers
27 views

Reference request -- distributions on Lie groups and enveloping algebras

I am looking for a reference detailing the convolution algebra of distributions on a Lie group supported at the identity and its isomorphism with the universal enveloping algebra of the associated Lie ...
szantag's user avatar
  • 111
2 votes
1 answer
92 views

Is there another way of deriving the Fourier transform of 1?

Reasoning with generalized functions we can derive that the Fourier transform of the Dirac delta function is: $$\langle \mathcal{F}[\delta(x)](p),\phi(p) \rangle=\langle \delta(x),\mathcal{F}[\phi(p)](...
Krum Kutsarov's user avatar

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