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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48 views

Solution of this definite integral?

I want to find the expression for the following integral $$\int_0^\infty\text{d}x\frac{e^{i k x}}{x}$$ I have tried deriving with respect to $k$, transforming into an integral over the whole real ...
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0answers
27 views

show that $f_{\epsilon} \in D(\Omega)$; moreover, $f_{\epsilon} \to f$ uniformly as $\epsilon \to 0$.

Let $K$ be a compact subset of $\Omega \subset \mathbb{R^m}$, $\Omega$ is open and nonempty and let $f \in C(\Omega)$ have support contained in $K$. For $\epsilon \gt 0$, let ...
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0answers
28 views

Fourier transform calculus of tempered distributions

For example I wanted to ask confirmation of this calculation, if $u \in \mathcal{S}'(\mathbb{R}^n)$ then $\widehat{D^\alpha u} =(2\pi i \xi)^\alpha \widehat{u}$. By definition $\langle \varphi , u ...
3
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1answer
22 views

Show that $f_{n}^2(x)$ does not converge in $D^1({\Omega})$

Let $$ f_n(x) = \left\{ \begin{array}{ll} n & \mbox{if $0 \lt x \lt \frac{1}{n}$};\\ 0 & \mbox{otherwise}.\end{array} \right. \ $$ I have to show that $\lim_{n \to ...
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0answers
20 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
26 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
16 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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2answers
43 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 ...
1
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1answer
18 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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0answers
57 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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1answer
25 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. ...
1
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1answer
52 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
18 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) ...
3
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0answers
55 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 ...
0
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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
30 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 ...
1
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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 ...
2
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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
24 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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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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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 ...
2
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1answer
78 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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2answers
115 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
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 ...
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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
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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 ...
3
votes
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 ...
2
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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 ...
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 ...
0
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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

Show that $T$ is independent of $\epsilon$

Define $$T: C_{0}^{\infty}(\mathbb{R})\to \mathbb{R}$$ by ...
2
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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 ...
1
vote
2answers
47 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 ...
1
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1answer
29 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 ...
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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 ...
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0answers
84 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 ...
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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.
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0answers
31 views

Silly question about sampling a function (distribution vs integral representation)

I have a stupid question... if i want to sample a continuous function i can use the Dirac delta distribution $$f(x_0) = \int_{-\infty}^{+\infty} \delta(x - x_0) f(x)dx,$$ which indeed involves the ...
0
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1answer
22 views

Doubts on distributional derivative: null derivative and non negative derivative

suppose we are in $\mathbb R^n$ and fix $E\subset \mathbb R^n$ and let $\chi_E$ be the indicator function of the set $E$. I'm reading an article that says that a vector field $X$ on $\mathbb R^n$, ...
3
votes
1answer
42 views

Traveling delta function $\delta(x - ct)$ as a distributional solution of the wave equation

I'm trying to show that a delta function $\delta(x - ct)$ is a distributional solution of the PDE $$ D_{(0,2)}u(x, t) = c^2 D_{(2,0)}u(x,t). $$ Here $D_{(i,j)}$ means $i$-th partial differentiation on ...
2
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2answers
49 views

Dirac Delta function at a point

From my understanding of the Dirac Delta function, it is infinitely thin and has a value of infinity at only a particular point. I also learned that $$\int_{-\infty}^{\infty} \delta(x-a) dx = 1$$ What ...
0
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1answer
48 views

Integral definition of delta function and Kronecker symbol

I know the following two definitions for the delta function and Kronecker delta, respectively: (1) $\int_{-\infty}^{\infty}\frac{e^{iwt}}{2\pi}\mathrm{d}t = \delta(w)$ (2) ...
5
votes
1answer
50 views

$5$ questions on the definition of the Gelfand triple

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb F\in\left\{\mathbb R,\mathbb C\right\}$, $\left\|\;\cdot\;\right\|$ be the norm induced by ...
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= ...
1
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1answer
29 views

Fourier transform of a distribution null on [-1,1]

Here is an interesting problem : Let $f \in \mathcal{C}^0 ( \mathbb{R})$ bounded, and $T_f \in \mathcal{S}'(\mathbb{R})$ defined by $\displaystyle \langle T_f, \phi \rangle = \int_{\mathbb{R}} ...
2
votes
1answer
68 views

What is meant by this notation for a sum of distributions

$\newcommand{ \reals }{ \mathbb{R}} $ $\newcommand{ \distfns }{ \mathcal{D}'(\reals) } $ $\newcommand{ \testfns }{ \mathcal{D}( \reals ) } $ $\newcommand{ \ints }{ \mathbb{Z}} $ $\newcommand{ \x }{ ...