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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10 views

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well.

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well. Let $f,g \in C(\Omega)$. Since $f\ne g$, there is a $x_0 \in \Omega$ such that $f(x_0)\ne g(x_0)$. Hence ...
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0answers
22 views

Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$

Let $f$ be a generalized function on $\Omega$. Let $G$ be an open subset of $\Omega$. Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$ Let $\phi ...
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0answers
6 views

What it means for a generalized function to be periodic or radially symmetric??

Let $T$ be a generalized function. I need to provide definitions for $T$ to be periodic and radially symmetric. A function (on $\mathbb{R})$is said to be periodic if there exists a $p \in ...
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1answer
51 views

Distribution of a function of a random variable

Suppose we have continuous random variable $X$ with distribution $f_X$. That is $$ P\left(a \le X \le b \right) = \int_{a}^{b} f_X(x) \ dx $$ Now suppose I have a function $\phi: \Bbb{R} ...
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1answer
173 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
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2answers
42 views

What is the definition of the order of a distribution?

A linear functional $T$ on $\mathcal{D}(\Omega)$ is a distribution if $\phi_n \to 0$ in $\mathcal{D}(\Omega)$ $\Rightarrow$ $T(\phi_n) \to 0$ in $\mathbb{R}$. But I cannot find what the order of a ...
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0answers
46 views

Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...
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0answers
48 views

Fourier distribution $\frac{e^{i|x|}}{|x|}$

I need help to calculate Fourier transform in distribution sense of $\frac{e^{i|x|}}{|x|}$ in $D'(\mathbb{R}^3)$ we have $ \frac{e^{i|x|}}{|x|} \in L^1_{loc}(\mathbb{R}^3)$ edit, Let ...
1
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1answer
17 views

Tempered representatives of a special class of distributions

Suppose that a distribution $R\in D'(\Bbb R)$ satisfies the following estimation for an independent constant $c$: $$\forall \phi\in D(\Bbb R)\quad |\langle R,\phi\rangle|\le c\|\phi, \,L^1(\Bbb ...
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1answer
13 views

Bump functions converging to an indicator

Suppose $K\subset\mathbb{R}^n$ has a smooth boundary, and let $\phi_s(x)$ be bump functions converging pointwise to the indicator of $K$, i.e. ...
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1answer
46 views

limit of a distribution

Show that $$\lim_{\epsilon \to 0^{+}} \left\langle \frac{\epsilon}{x^{2}+ \epsilon} ,\phi \right\rangle =\langle \delta,\phi\rangle $$ where $\phi\in D(\mathbb{R}) $ and $ \frac{\epsilon}{x^{2}+ ...
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0answers
17 views

convolution properties of distributions

Let $f,g,h \in D'(R^n)$. How we define the convolution of these functions? I'm trying to show some properties of convolutions such as $\delta\ast f=f$ $(f\ast g)' = f'\ast g=f\ast g'$ $(f\ast g) ...
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1answer
465 views

Convolution of distributions.

$\newcommand{\supp}{\operatorname{supp}}$ We are given with distributions $f,g \in D'(\Bbb R)$. If $\supp f\subset (-\infty,a)$ and $\supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined ...
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1answer
45 views

Borel lemma : wikipedia proof

In the proof of Borel's lemma, I don't understand why we use $\psi\left(\frac{t}{\epsilon_m}\right)$ for a sufficient small $\epsilon_m$ and not $\psi(t\cdot \epsilon_m)$, as you need to keep ...
2
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2answers
28 views

How can we prove that the generalized stochastic process induced by a real-valued Brownian motion is Gaussian?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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0answers
13 views

Covariance functional of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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1answer
20 views

Prove order of distribution $\Lambda_{1/x}$ is 1

The distribution is defined as: $$\Lambda_{1/x}(\varphi)=\lim_{\varepsilon\rightarrow0+}\int_{\mathbb{R}\backslash(-\varepsilon,\varepsilon)}\frac{\varphi(x)}{x}\ \mathrm{d}x$$ I tried integrating ...
3
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1answer
48 views

Computing the limit of an integral

Consider the following integral $$ \int_{-\infty}^{\infty}f(t) K(\frac{a-t}{h})dt $$ where (1) $h>0$, $a \in \mathbb{R}$ (2) $f:\mathbb{R}\rightarrow[0,\infty)$ is such that ...
1
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1answer
134 views

Reference for “distributional derivative being zero implies being constant”

I know that if a distribution (generalized function) has zero derivative, then it is a constant. I also know the proof. But I have a hard time finding a reference which contains a statement of this ...
4
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1answer
97 views

Limit of distribution

Let $T\in\mathcal{D}'(\mathbb{R})$ be a distribution on the set of smooth functions of compact support $\mathcal{D}(\mathbb{R})$ such that $$ \forall_{g\in\mathcal{D}(\mathbb{R})}~|\langle T, g ...
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2answers
22 views

functions acting as linear functionals on their dual space

Supposing $f\in L^p$, where p and q are conjugate exponents, what does it mean that "f is completely determined by its action as a linear functional on $L^q$"? (Quoting Folland's Real Analysis here). ...
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1answer
34 views

Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
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1answer
23 views

What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
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1answer
21 views

How to show that $\delta^3(\vec{x}-\vec{a})=\lim_{\alpha\to0} \frac{\alpha/\pi^2}{(\alpha^2+|\vec{x}-\vec{a}|^2)^2}$

I hope to show that: $$\delta^3(\vec{x}-\vec{a})=\lim_{\alpha\to0} \frac{\alpha/\pi^2}{(\alpha^2+|\vec{x}-\vec{a}|^2)^2}$$ I want to show by: $$\int^{+\infty}_{-\infty} f(\vec{x}) \lim_{\alpha\to0} ...
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1answer
31 views

Does the weak divergence exist for each $\mathcal L^2(\Omega;\mathbb R^d)$-function?

Let $\Omega\subseteq\mathbb R^d$ be open. $v:\Omega\to\mathbb R$ is called weak divergence of $u:\Omega\to\mathbb R^d$ $:\Leftrightarrow$ $$\int_\Omega v\varphi\;{\rm d}\lambda=-\int_\Omega\langle ...
2
votes
1answer
70 views

$\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$ with distributions defined on Schwartz space

I know, from recent enlightening answers received here, that, if we define the distribution represented by Dirac's $\delta$ on the space $K$ of test functions of class $C^\infty$ whose support is ...
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0answers
13 views

Why can “sufficiently smooth” distributions of $C_c^\infty([0,\infty)\times G)$ be represented as functions of $t\in [0,\infty)$ and $x\in G$?

Let $G\subseteq\mathbb R^d$ be a bounded domain and $$\mathcal D:=C_c^\infty([0,\infty)\times G)\;.$$ Assuming that $\mathcal D$ is equipped with the usual locally convex topology, the space of ...
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1answer
39 views

What's the distributional derivative of a Banach space valued almost surely continuous stochastic process?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\lambda$ be the Lebesgue measure on $[0,\infty)$ $(H,\left\|\;\cdot\;\right\|)$ be a Banach space over the field $\mathbb ...
6
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2answers
102 views

Dirac's delta in 3 dimensions: proof of $\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$

If $T_f$ is a distribution, i.e. a linear functional, continuous according to the convergence defined here, defined on the space $K$ of the functions of class $C^\infty$ that are null outside a ...
2
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1answer
50 views

Prove that $\frac{t}{t^2-1}$ is a tempered distribution

I want to compute the Fourier transform of $\frac{t}{t^2-1}$, and in order to do so I need to prove in which space is the function. Clearly the function is not $L^1(\mathbb{R})$ neither ...
5
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1answer
127 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
3
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2answers
50 views

Why is $\frac{d^2}{dx^2} e^{-|x|} = e^{-|x|} - 2 \delta(x)$

How can you show that $$\frac{d^2}{dx^2} e^{-|x|} = e^{-|x|} - 2 \delta(x) ? $$ I found this result using Wolfram Alpha and it seems strage to me, how the delta function appears here ...
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2answers
24 views

Why does convolution of delta function commute (test distribution perspective)?

If I understand correctly, for test functions $f$ we define $$ \langle\delta, f\rangle = f(0)$$ and convolution is defined as $$ \langle g * T, f\rangle = \langle T, g^- * f\rangle,$$ where $f$ ...
1
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1answer
39 views

(Distributional) Fourier transform

I need to calculate the (distributional) Fourier transform of $$ f(x) = \frac{x^2}{x^2+1}. $$ I unsuccessfully tried to find a differential equation for $f$ in order to solve the Fourier-transformed ...
11
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2answers
279 views

Oscillatory integral giving me the willies

So now that my term's over, I've been brushing up on my quantum field theory, and I came across the following line in my textbook without any justification: ...
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0answers
82 views

Characterize a set of functions

While computing matrix elements of the evolution operator in Quantum Field Theory for the harmonic oscillator using the path integral formalism, I came across the assumption that all physically ...
2
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1answer
28 views

Push forward of the Lebesgue measure is the Haar measure of the Carnot group

I have the following problem. I have a Carnot group $(\mathbb G,*)$ which is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ is stratified as $\mathfrak g= ...
2
votes
2answers
49 views

Limit of $\frac{1}{x+i y}$ for $y \rightarrow 0$ and distributional relations

So I know for $y \rightarrow 0$ I have the following (distrubutional) relation: $\frac{1}{x+i y} = \frac{x}{x^2+y^2} - i \frac{y}{x^2+y^2} = P(1/x) - i \pi \delta(x) $ where in the last expression ...
0
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1answer
30 views

What is the divergence $\operatorname{div}u$ of a $L^2(\Omega)^d$ function $u$?

Let $\Omega\subseteq\mathbb R^d$ be open. In the book Navier-Stokes Equations - Theory and Numerical Analysis by Roger Temam the author is using the divergence $\operatorname{div}u$ of a ...
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1answer
26 views

What is the gradient of a distribution?

Let $\Omega\subseteq\mathbb R^d$ be open and $\mathcal D(\Omega)$ be the set of $C^\infty(\Omega)$-functions with compact support equipped with a locally convex topology. Let $\mathcal D(\Omega)'$ ...
2
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1answer
17 views
1
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2answers
46 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 ...
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1answer
27 views

Verifying equivalent definitions of continuity for distributions

For an arbitrary vector space $V$ over $\mathbb{F}$, consider continuous linear maps $f: V \to \mathbb{F}$ where continuity is defined as sequential continuity, i.e. if $\phi_j \to \phi$ in $V$ then ...
5
votes
2answers
101 views

What is the Fourier transform of $\exp(2 \pi i / x)$?

The Fourier transform of $e^{2 \pi i / x}$ makes sense as a distribution, I believe. Does it have a nice expression in terms of functions and well-known distributions (e.g. Dirac delta)?
3
votes
1answer
97 views

Verifying that $\lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x)$

I'd like to show that: $$ \lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x). $$
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1answer
113 views

I want to prove $f\notin W^{1,1}(\mathbb{R},\gamma_{1})$

Let $\gamma_{1}=\mathscr{N}(0,I_{1})$ in $\mathbb{R}$ be the standard Gaussian measure. Consider the sequence $(f_{n})_{n\in\mathbb{N}}\in C_{b}^{1}(\mathbb{R})$ defined by $$f_{n}(x)=\begin{cases} ...
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0answers
22 views

Convolution of delta-ish functions

I would like to compute the convolution of a function with itself, where the function is $f(x) = \frac{\delta(x)}{x}$. When there is a shift in the delta function it is easy to compute, but this one ...
0
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2answers
77 views

Show that the limit of distributions is Dirac delta

I would like to show that the following statement is true $ \lim_{a\searrow 0} \theta(x)\frac{x^{1-a}}{\Gamma(a)} = \delta(x). $ $\Gamma$ is the gamma function. The above limit is in the sense of ...
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4answers
3k views

How to prove that the derivative of Heaviside's unit step function is the Dirac delta?

Here is a problem from Griffith's book Introduction to E&M. Let $\theta(x)$ be the step function $$\theta = \begin{cases} 0, & x \le 0, \\ 1, & x \gt 0. \end{cases} $$ The ...
1
vote
1answer
37 views

prove $\frac{n e^{-n^2 x^2}}{\sqrt{\pi }}$ converges to $\delta(x)$

How can show as $n$ goes toward infinity the sequence converges to $\delta(x)$ my problem is I don't know how to show this.