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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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0answers
45 views

Why CDF is not coming to 1? [duplicate]

Suppose, $F_X(x)=-\frac{x}{a^2}+\frac{2\sqrt{x}}{a}$ And, $f_X(x)=\frac{d}{dx}F_X(x)=\frac{1}{a\sqrt{x}}-\frac{1}{a^2}$, Here, $0\leq x \leq a^2$ Similar, $f_Y(y)=\frac{1}{a\sqrt{y}}-\frac{1}{a^2}$, ...
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2answers
78 views

Why CDF is not coming to 1, when i put the maximum range?

Suppose, $F_X(x)=-\frac{x}{a^2}+\frac{2\sqrt{x}}{a}$ And, $f_X(x)=\frac{d}{dx}F_X(x)=\frac{1}{a\sqrt{x}}-\frac{1}{a^2}$, Here, $0\leq x \leq a^2$ Similar, $f_Y(y)=\frac{1}{a\sqrt{y}}-\frac{1}{a^2}$, ...
0
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0answers
47 views

Distributional solution.

How to solve the following differential equation on the space of distributions? $$ u'+tu=K(0,1), $$ Here $K_{(0,1)}$ is the characteristic function of the closed interval between $0$ and $1$. I ...
2
votes
0answers
32 views

Limit of function scaling when parameter tends to zero

Given any $\lambda>0$, consider the dilation operator $D_{\lambda}f(x)=f(\lambda x)$. Does it make any sense to consider the limit $\lambda\rightarrow 0$? Given that the function spaces I am ...
1
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1answer
54 views

Derivative of ln|x| is the principal value of 1/x. Distribution Theory.

I have been looking at the proof for $\frac{d}{dx}\ln|x|=p.v.(\frac{1}{x})$ in the context of distributions and I am having trouble understanding why in the second term after integration by parts the ...
0
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0answers
23 views

How to distribute N alike objects to M people with given constraints

The value of each object is different for every person. For eg- There are three people A,B and C and the value of each object for them are 3,4 and 5 respectively. We are also given the net value we ...
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0answers
10 views

Existence of a smooth compactly supported primitive [closed]

Under what conditions is a compactly supported smooth function the partial derivative of another compactly supported smooth function?
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1answer
35 views

The probability density function $f(x)$ of a random variable $X$ is symmetric about $0$. Then we have:

The pdf $f(x)$ of a random variable $X$ is symmetric about $0$. Then $\int_{-2}^{2}\int_{-\infty}^{x}f(u)dudx= $? My input: $\int_{-2}^{2}\int_{-\infty}^{x}f(u)dudx$ $\int_{-2}^{2}F(x)dx\ $ This ...
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1answer
23 views

How to show that the cdf of Y is $F(\sqrt{Y})$ if cdf of $X$ is $F(x)$, and $Y=X^2$?

I was trying to work through how to find the CDF of $Y$ if we know CDF of $X$ is $F_X(x)$, and $Y=X^2$. So if I plug in $y=x^2$ then I can intuitively get $$F_Y(y)=+\sqrt y$$ I solved this simply ...
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0answers
14 views

Reference Sobolev and distribution

I've studied distribution in dimension 1 seriously (maybe 20 hours). And recently we've learnt Sobolev space in dimension 1 in the context of distribution. I would like to learn more about it. I ...
2
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0answers
24 views

C^infty approximation of Heaviside step function: definite integral of a bump function

I'm wondering if there is an analytical form for a $C^\infty$ approximation with a compact support of a Heaviside step function $f(x) = I_{x \geq 0}$. In attempting to construct one, I'm taking a bump ...
2
votes
1answer
51 views

How can we prove the scaling property of the Dirac delta function rigorously?

Let $(\Omega,\mathcal A)$ be a measurable space $\omega\in\Omega$ $\delta_\omega$ denote the Dirac measureat $\omega$ on $(\Omega,\mathcal A)$ $E$ be a $\mathbb R$-Banach space $\mathcal M$ denote ...
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0answers
17 views

Boundary functions of elements in $H^\infty(U)$

Let $U$ the upper-half plane and let $H^\infty(U)$ the set of bounded holomorphic functions defined on $U$. From the theory of harmonic bounded functions in the upper-half plane, we know that: $$\...
2
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1answer
35 views

How do we need to read the Dirac-comb identity?

Let $(\Omega,\mathcal A)$ be a measurable space $\omega\in\Omega$ $\delta_\omega$ denote the Dirac measureat $\omega$ on $(\Omega,\mathcal A)$ $E$ be a $\mathbb R$-Banach space $\mathcal M$ denote ...
1
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1answer
25 views

Existence of a smooth extension vanishes at infinity

in my research I saw this argument used in an article and I don't know if it's true (I believe yes). Let $\Omega \subset \mathbb{R}^3$ be an open, bounded and simply connected domain and let $\...
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0answers
40 views

Integral of Dirac-delta squared

I read this Phys.SE post on integrating $\delta (t)^2$, and the top answer concludes that it results in $\delta(\lambda-\lambda)$. Shouldn't this mean that it results in $\delta(0) $, which is ...
0
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1answer
34 views

A uniqueness problem of PDE

Let $P(D)$ be a nontrivial constant coefficient partial differential operator, where $D:=-i\partial \,.$ Show that the equation $P(D)u=0$ has no nonzero solutions $u\in M^\prime$, where $M^\prime$ ...
0
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0answers
15 views

Quantifying the spread of a ratio distribution

My question is similar to but more specific than this one. In short, how do I quantify the spread of a ratio distribution when the ratio of averages is the better method of quantifying the mean ...
3
votes
1answer
111 views

$L^p(\mathbb{R})$ convergence of Fourier truncated integral to $f$ for $f\in L^p(\mathbb{R})$ and $p\in(2,+\infty)$.

From the Hilbert transform theory it follows that $$\forall p\in(1,2], \forall f\in L^p(\mathbb{R}), \left\|f-\left(x\mapsto\int_{-M}^M\hat{f}(\xi)e^{2\pi i\xi x}\operatorname{d}\xi \right)\right\|_{L^...
1
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0answers
50 views

Distributional proof of Helmholtz decomposition

Wikipedia page on Helmholtz decomposition, https://en.wikipedia.org/wiki/Helmholtz_decomposition presents an elegant proof which makes use of the fundamental solution/Green function of the Laplace ...
0
votes
1answer
41 views

Introduction to the Theory of Distributions, Friedlander and Joshi, Exercise 2.2

The (first part of the) problem reads: Show that $$f_t(x)=\frac{\sin(xt)}{\pi x}\to\delta \ \text{in} \ \mathcal{D}'(\mathbb{R}) \ \text{as} \ t\to\infty.$$ If I understand correctly, this means ...
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1answer
33 views

Introduction to the Theory of Distributions, Friedlander and Joshi, Exercise 2.1

The problem reads: Show that $$\frac{1}{\pi}\frac{\epsilon}{x^2+\epsilon^2}\to\delta \ \text{in} \ \mathcal{D}'(\mathbb{R}) \ \text{as} \ \epsilon\to 0^+.$$ If I understand correctly, this means ...
1
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0answers
24 views

A limit that exists strongly in Sobolev space $H^s$ and weakly in $H^{s+1}$

In his paper 'Nonstationary flows of viscous and ideal fluids in $\Bbb{R}^3$' Kato mentioned the following: a limit that exists strongly in $H^{m-1}$ and weakly in $H^{m}$ what is explicitly the ...
2
votes
1answer
38 views

Partial Derivative of Multivariate Indicator Function in the Distributional Sense

Define $f:\mathbb{R}^2\to\mathbb{R}$ by $\left(x,y\right)\mapsto1$ if $x^2+y^2\leq1$ and $\left(x,y\right)\mapsto0$ otherwise. Is there a definition of $\partial f/\partial x$ in the distributional ...
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0answers
19 views

non negative distributional derivative

Let $J$ be an open interval and let $f:J\to\mathbb{R}$ be locally integrable. Suppose that for all $0\leq\theta\in C^\infty_c(J)$ it holds $$ \int_J f(t)\,\theta'(t)\,dt\geq 0\,, $$ namely the first ...
1
vote
1answer
22 views

Show that if $F$ is a distribution, such that $|\xi|^{2}F=0$, then support (F) is subset of $\{0\}$.

Show that if $F$ is a distribution, such that $|\xi|^{2}F=0$, then support (F) is subset of $\{0\}$. My approach: Recall that, the support of a distribution $F$ is the complement of the union of all ...
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0answers
46 views

Reed and Simon definition of product of distributions

Let $\mathcal{D}$ denote the space of $C^{\infty}$, compactly supported functions on $\mathbb{R}^{d}$, and let $\mathcal{D}'$ denote its dual (i.e. the space of distributions). In volume II of Reed ...
4
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0answers
41 views

Uniform convergence of convolution of a distribution with a test function

For an exercise I have to show the following: Let $u_j \to u$ in $\mathcal{D'(\mathbb{R}^n)}$ and let $\phi_j \to \phi$ in $C^{\infty}_0(\mathbb{R}^n)$. Show that $$ \lim_{j\to \infty} u_j * \phi_j ...
4
votes
3answers
117 views

How to prove $\lim_{t\rightarrow \infty} \frac{e^{i x t}}{x- {i\mkern1mu} 0^+} =2 \pi {i\mkern1mu} \delta(x)$

Recently I meet following two equations in physics. Their proof is like a magic and I can't understand. $$\lim_{t\rightarrow -\infty} \frac{e^{i x t}}{x- {i\mkern1mu} 0^+} =0$$ $$\lim_{t\rightarrow \...
2
votes
1answer
43 views

Dirac delta uder coordinate transformation

I am having some issues understanding Dirac's delta function/distribution behaviour under change of coordinates. There is a statement, if $(x_1,\ldots,x_n)$ are cartesian coordinates and $y_1,\ldots,...
2
votes
2answers
70 views

Fourier transform of $1/x^2$ given by Mathematica

Mathematica gives $-k \sqrt{\frac{\pi}{2}} \text{sgn}(k)$ as the Fourier transform of $1/x^2$ (i.e., the result of the command FourierTransform[1/x^2, x, k]). And ...
1
vote
3answers
41 views

Does convergence in $D'(\Omega)$ imply convergence in $L^2(\Omega)$?

Consider $\Omega \subset \mathbb R^n$ open. Let $\{f_n\}_{n\geq 1}$ and $f$ belong to $L^2(\Omega)$. Suppose that $f_n \xrightarrow{n\to\infty} f$ in $D'(\Omega)$, i.e., $\int_\Omega f_n \varphi \...
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0answers
7 views

Solution operator to scalar hyperbolic equation on Sobolev space $H^s$

If i have the solution operator $e^{itA}$ to the following equation $$\frac{du}{dt}=iA(x,D)u$$ where $A\in OPS^1$ is a peudo differential operator of order 1. Then how could one show that the solution ...
1
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1answer
43 views

Simplify exponential expression

Consider an exponential model with density $\theta e^{-\theta x}$ with $x> 0$ and $\theta >0 $. Derive LR test of approximate level $\alpha$ (For large sample size) for the hypothesis problem $...
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2answers
52 views

Confusion on delta function

How should one deal with $\delta(x-y)\partial_x[f(2x+y)\delta(x+y)]$? In other words, what is \begin{equation} \int\phi(x,y)\delta(x-y)\partial_x[f(2x+y)\delta(x+y)] dxdy ? \end{equation} My attempt ...
1
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1answer
72 views

Introduction to the Theory of Distributions, Friedlander and Joshi, Exercises 1.3 and 1.4

The problems are: Show that for each $\phi \in C_{C}^{\infty}(\mathbb{R})$, the principal value integral $$\text{p.v.}\int\frac{\phi(x)}{x}\text{d}x = \lim_{\epsilon \to 0^+}(\int_{(-\infty,-\epsilon)...
4
votes
2answers
136 views

How to solve $u_t + uu_x =\delta(x)$

I am studying from old exams and there is a problem which is traffic flow with a ramp. I have never seen this type of problem in class, so for the simple case, how would I solve $$u_t + uu_x =\delta(...
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2answers
44 views

Cours of convolution product in theory of distribution

i search a good and complet cours of convolution product in theory of distribution who contains definition of convolution of an function with distribution and convolution of two distributions (for ...
4
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1answer
50 views

Differentials of non-smooth functions, wedge products of currents?

In a paper of McMullen he considers foliations on a manifold determined by a closed 1-form $\rho$. He says an $L^\infty$ function $f$ is constant on the leaves of the foliation if "$df \wedge \rho = ...
0
votes
1answer
36 views

Dirac Delta in polar coordinates, integrating delta from 0 to $\infty$

Many sources mention how to recast Dirac's Delta function from Cartesian into polar or spherical coordinates. They say e.g. for polar: $$\int_{-\infty}^{\infty}\delta(\vec{x})\,{\rm d}^3x=\int_{-\...
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1answer
43 views

How to understand partial differential equations in the sense of distribution?

I have just studied some elementary distribution theory. However, when attempting to apply them in solving partial differential equations I encounter the following confusion. Consider the heat ...
0
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0answers
56 views

Improper integral ${\cal P}\int_{-\infty}^{\infty}{1\over x} {\rm d}x$

I am wondering about the principal value in the improper integral ${\cal P}\int_{-\infty}^{\infty}{1\over x} {\rm d}x$. Posts like: Principale value, how can we consider it? explain that the ...
2
votes
2answers
83 views

Identifying $\int_{-\infty}^\infty e^{i k x} dx$ as Dirac delta distribution

The expression $\int_{-\infty}^\infty e^{i k x} dx$ is sometimes identified as the Dirac delta function. This identification is said "formal" or "symbolic", and some physics texts say that the theory ...
1
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0answers
42 views

Convolution Theorem for Distributions

I am searching for a version of the convolution theorem for functions (i.e. $\hat{f \cdot g} = \hat f \star \hat g$) that also applies to Distributions/tempered Distributions. Is it automatically true,...
4
votes
2answers
90 views

Laplacian in 4-dimensions

I know for 3-D $$\nabla^2 \left(\frac1r\right)=-4\pi\, \delta(\vec{r})\,.$$ I would like to know, what is $$\text{Div}\cdot\text{Grad}\left(\frac{1}{r^2}\right)$$ in 4-Dimensions ($r^2=x_1^2+x_2^2+x_3^...
1
vote
0answers
55 views

Fourier transform of the projection operator in 3D

I have a vector field $A_i({\bf r})$, a Fourier transform given by $$ \tilde A_i({\bf k}) = \int d^3 r~e^{i {\bf r.k}}A_i({\bf r}),$$ and projections given by $\mathcal P_{ij}(\hat r) = \delta_{ij}...
0
votes
1answer
39 views

Confusion over definition of distributional derivatives

Suppose I want to differentiate the following integral pairing with respect to some scalar $s\in \mathbb{R}$: $\int \psi(x,s)\rho(x,s)d\nu(x)$. Here $\psi \in C^\infty (\mathbb{R}^n\times \mathbb{R})$,...
1
vote
2answers
43 views

How does $\int (u*v)dx = \int u dx \int v dx $ follow from $ (u*v)*w = u*(v*w) $ by taking $w = 1$?

In Hormander's first book on PDEs he states on page 17 that $$ (u*v)*w = u*(v*w) $$ if all except one of the continuous functions $u,v,w$ $\in C(\mathbb{R})$ have compact support. He then says ...
1
vote
0answers
47 views

Contour Integral of Geometric Series

I'm uncertain about the subtleties of the following contour integration, so maybe somebody can tell me what is precisely going on. First consider the function $$ f(z)=\sum_{n=0}^\infty z^n = \frac{1}{...
0
votes
0answers
43 views

Expression for the density function of a smooth function

I am working on tomographic methods in which the data is the "distribution" of values along a line rather than an integral. Given a measurable function $f:[0,1] \rightarrow \mathbb{R}$ one can define ...