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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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1answer
23 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
26 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 ...
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0answers
52 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
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2answers
70 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 ...
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0answers
31 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,...
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0answers
67 views

Continuous distribution functions getting uniformly distributed on $(0,1)$ [closed]

Let ($\Omega, \mathcal{F}, \mathbb{P}$) be a probability space, ($X_i)_i$, $X_i:\Omega\mapsto\mathbb{R}$ a set of random variables with continuous distribution functions $(F_i)_i$. (i.e. $F_i(x)=\...
3
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1answer
59 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^...
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0answers
48 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}...
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1answer
35 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
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2answers
30 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 ...
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0answers
42 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}{...
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0answers
40 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 ...
1
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2answers
74 views

Distribution theory - Find a solution to the linear partial differential equation $\partial u+au=\phi$

This was a problem on my final last semester. I am trying to learn how this is done for the future. I thought that I figured it out and I emailed the instructor and he said there was a simpler ...
1
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1answer
20 views

Source of non-linear Laplace equation

Consider the following non-linear generalization of the Laplace equation $$\Delta \phi - \frac{\sum_i (\partial_i \phi)^2}{2 \phi} = 0$$ I am looking for spherically symmetric solutions, so I assume ...
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3answers
39 views

If X is a non-negative continuous random variable, show that $E[X]=\int_0^\infty (1-F(x)) dx$

There is a hint to solve this by using integration by parts. So I have $$ u = x \space \mbox{then} \space u'=1$$ $$v' = F(x) -1 \space \mbox{then} \space = ? $$ QUESTION: What is $v'$ ? What is ...
1
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1answer
29 views

Prove $\lim_{t\to\infty} \sin(tx) \text{P.V.}\frac{1}{x} = \pi \delta$ in the distributional sense

Prove $\lim_{t\to\infty} \sin(tx) \text{P.V.}\frac{1}{x} = \pi \delta$ in the distributional sense. This was the work I had put into the question: For $\phi \in \mathcal{D}(\mathbb{R})$, \begin{...
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0answers
26 views

Radon-Nikodym derivative when distributions are involved.

I have two cases I'm interested in looking at (I'm computing two KL divergences specifically). The domain is $\Omega \in \mathbb{R}^{2n}$. Let's say $(x, y) \in \mathbb{R}^{2n}$. Assume the ...
1
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2answers
40 views

If $\phi \in C_c^{\infty}(\mathbb{R})$ then$ \frac{\phi}{x}\in L^1(\mathbb{R})$?

If $\phi \in C_c^{\infty}(\mathbb{R})$ then$ \frac{\phi}{x}\in L^1(\mathbb{R})$ ? Let $\phi$ be a smooth, compactly supported function and $\frac{1}{x}$ be an odd function defined on $\mathbb{R}$. ...
2
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1answer
64 views

Is it possible to extend the notion of $H$-convergence to the case of distributions?

The usual $H$-convergence is defined for operators of the following form (for the sake of simplicity, restrict ourselves with the one-dimensional case): $$ \frac{d}{d x}\left[A_\varepsilon(x) \frac{d ...
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0answers
44 views

Evaluating $I = \lim_{\varepsilon \to 0} \int_{-\varepsilon}^{\varepsilon} \frac{1}{\varepsilon} u(x) dx$ through a change of variables?

I want to evaluate the following integral using change of variables but I'm not sure if what I'm doing is correct. It 'feels' like I should get the answer $I = 2 u(0)$. $$ I = \lim_{\varepsilon \to 0}...
2
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1answer
47 views

does weak derivative zero imply independence?

If $f\colon\mathbb{R}^2 \to \mathbb{R}$ is a continuous function, and $\partial_x f = 0$ in the weak sense, i.e. $$ \iint f(x,y) \partial_x \phi( x,y) dx dy =0 $$ for all $\phi \in C^\infty_c(\mathbb{...
2
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0answers
41 views

Fourier transformation on $\{T_\theta\}$ is surjective

Let $S$ Schwartz space on $\mathbb{R}^2$. For $f\in S$, let $T(f):=\int _{-\infty}^{\infty} f(0,y)dy$ and $T_{\theta} (f):=T(f(x \cos{\theta} -y \sin{\theta}, x\sin{\theta} +y\cos{\theta}))$ and we ...
1
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1answer
33 views

Inverse Fourier Transform of the Hdamard's finite part - 2 dimensions

I'm trying to compute the Fourier transform of the Hadamard's finite part distribution in two dimensions, and it's giving me a little trouble. In two dimensions, we defined the finite part $\text{PF}\...
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0answers
32 views

(notation) How to denote pointwise product in first variable and tensor product in second

Context/Question: While trying to formulate a question involving a product of bi-distributions, I realized I don't know the relevant notation to precisely state what I'm trying to do. Specifically, ...
3
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1answer
42 views

Rigorous derivation of the long-time limit of oscillatory integrals

I am trying to estimate the following integrals in the limit $t\to+\infty$: $\displaystyle\int_{-\infty}^{+\infty}\mathrm d\omega\,f(\omega)\frac{1-\cos(\omega t)}{\omega^2}$ and $\displaystyle\int_{-...
2
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2answers
37 views

Why $\frac{\varepsilon}{x^2+\varepsilon^2}$ converges in the sense of distributions to a constant times the Dirac delta

The integral of $f_\varepsilon(x)=\frac\varepsilon{x^2+\varepsilon^2}$ is the tan inverse, which is well behaved anywhere on $\mathbb{R}$, and so $f_\varepsilon$ is in $L^1_\text{loc}(\mathbb{R})$. ...
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0answers
23 views

Convergence of Parte Finie of $x \mapsto \frac{1}{x^2}$

I am working through some distribution theory notes, and was specifically working on this example Derivative of principal value distribution $1/x^2$ is equal to finite part distribution $-1/x^2$?. ...
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0answers
29 views

Applying the generalised Leibniz rule multi-index form

The multi-index Leibniz rule states $\partial^{\alpha}(fg) = \sum_{\beta \leq \alpha}{\alpha \choose \beta} (\partial^{\beta}f)(\partial^{\alpha - \beta}g)$ Where $ \alpha, \beta$ are multi-indices. ...
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1answer
22 views

Distribution Theory injective mapping proof

The question is: $<g, \Phi>=0, \forall \Phi \in C_{c}^{\infty}(X), \text{then } f=0.$ And the map $L_{loc}^{1} \rightarrow D'(X)$ where $D'(X)$ denotes the set of distributions on $X$, and $X \...
1
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1answer
48 views

Proof Assistance / Verification: Distribution Theory

The question is (1): Prove that the support of $ \delta = \{0\} $ where $ \delta $ denotes the dirac-delta distribution. (2): Show that if $$ f \in C(\mathbb{R^{n}}) $$ is a function then its ...
1
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1answer
30 views

Use Dominated Convergence theorem in Schwartz convergent sequence

Let $\{f_k\}_{n \in \mathbb N} \subset S(\mathbb R^n)$ converging to $f \in S(\mathbb R^n)$ in the $S(\mathbb R^n)$ topology. Is it true that $\int_{\mathbb R^n} f_k$ converges to $\int_{\mathbb R^n} ...
2
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0answers
48 views

Proof related to tensor of distributions properties

I want to prove the following result Proposition 1 Let $T \in D'(\mathbb R^n)$ and $S \in D'(\mathbb R^m)$. Then $$ \langle T_x, \langle S_y, \varphi(x,y) \rangle \rangle= \langle S_y \...
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0answers
71 views

Doubts about proof on distributions theorem

Recently, I've been reading Notes of Thierry Ramond about distributions theory and I arrived to the following result (Page 72), Proposition 4.2.3: Let $\Omega \subset \mathbb R^p$ be an open ...
2
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1answer
38 views

Extension of a distribution

Let $\Omega \subseteq \mathbb R^n$ be an open convex set and $T \in D'(\Omega)$. Is it possible to extend $T$ to a distribution in $D'(\mathbb R^n)$? I've been looking in some texts (Rudin and this ...
2
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1answer
28 views

Embedding $L^p(\mathbb{T})$ into $h(D)$

Let $\mathbb{T}$ be the torus and $D$ the unit disk. Let $h(D)$ be the space of the harmonic complex functions in the unit disk equipped with its natural topology, i.e. the topology of the uniform ...
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0answers
40 views

Integral of function composed with two Dirac deltas

I'm looking to evaluate the following integral: $\int_{- \infty}^{\infty} f(x) \delta(x-a) \delta(x-a)dx$ If instead of two identical Dirac deltas we had: $\int_{- \infty}^{\infty} f(x) \delta(x-a)...
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0answers
14 views

How do I find $Q^{-1}$ tail distribution function on nomograph?

How can I get the value of inverse $Q$ function (tail distribution function of the standard normal distribution) only using this nomograph, without any calculator or software? How do i find for ...
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0answers
24 views

distributional derivative of characteristic function of rational numbers.

I know that the Heaviside function ($H(x) = 1$ if $x\ge 0$ and $0$ if $x<0$) has derivative $\delta_o$ on $(-1,1)$ (in sense of distribution). Now I want to find the distributional derivative of $\...
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1answer
17 views

Finding the solution for an operator in an equation

Problem Find the fundamental solution about the operator $$z=\frac{d3}{dx^3}-7 \frac{d}{dx}+6$$ and check if it's true Solution $y'''=-7y'(x)+6y(x)=0$ $y(0)=0$ $y'(0)=0$ $y''(0)=1$ The ...
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1answer
28 views

CDF for exponential family

Might be a stupid question though... we know for Exponential Family, we have the density is $$f(x|\theta)=h(x)\exp[\eta(\theta)\cdot T(x)-A(\theta)]$$ Is there a general form for the CDF function? I ...
3
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1answer
40 views

An example of a non-convergent Cauchy sequence in $C^\infty$ with the Sobolev norm

I apologize in advance if this question is a duplicate, but I couldn't find an answer. Studying Sobolev Spaces, I came across with the following proposition: The Sobolev Space $H^m(\Omega)$ is the ...
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1answer
33 views

Functions sequence derivative and Fourier transform analysis through distributions

Let $f_n : \mathbb{R} \to \mathbb{C}$ $$\begin{align}&f_n(x) = n \sin n \pi x \quad 0 \leq x \leq 1 \\ &f_n(x) = 0 \quad x \lt 0 \lor x \gt 1\end{align}$$ Can you calculate Fourier transforms ...
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0answers
35 views

How to show that a function defines a tempered distribution and compute the Fourier transform

Consider the following functions : $f(x) = 1 + x \cos(3x)$ and $g(x) = x \sin^2(x)$. How can I show that $f$ and $g$ define tempered distributions on $\mathbb{R}$ and compute their Fourier transforms (...
2
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0answers
41 views

Distributional derivative only generalization?

Suppose $H_1$ and $H_2$ are Hilbert spaces which lie between test-functions and distributions, i.e. $$ \mathcal{D}(\mathbb{R}) \subset H_i \subset\mathcal{D}'(\mathbb{R}), $$ such that the inclusions ...
2
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1answer
73 views

What is the integral of $\int_{-\infty}^{+\infty} H(t)\delta(t)dt$ ($H(t)$ Heaviside step, $\delta(t)$ Dirac delta)?

I was trying to figure out what is the integral of $$\int_{-\infty}^{+\infty} H(t)\delta(t)dt,$$ where $H(t)$ is the Heaviside step and $\delta(t)$ is the Dirac delta. A first approach: We ...
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0answers
24 views

Integration of representations of a locally compact group

I am studying Alain Robert's Introduction to the Representation Theory of Compact and Locally Compact Groups, where I found a notion that might be generalized. Given a locally compact group $G$ and a ...
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0answers
32 views

Structure of tempered distribution

I was reading a note on distributions. The author left the following Schwartz representation theorem as an exercise: I'm trying to prove the theorem. According to the hints, I've done all the ...
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2answers
25 views

Hyperbolic tangent expansion in temperature

I'm struggling with the following type of integral $$ \int \limits_0^\infty \mathrm{d} x \, f(x) \tanh \frac{x}{T} $$ I'm desperately trying to somehow "expand" the hyperbolic tangent for low ...
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0answers
13 views

Inverse image of a product fonction x distribution

Let $f : M \to N$ be a submersion between smooth manifold. Let $u$ be a Schwarz distribution on $N$ and $g: N \to \mathbb{R}$ a $C_{\infty}$-function. What can we say about $f^{*}(g.u)$ ? Is it the ...
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0answers
24 views

Wavefront Set of the Heaviside Function

If we define the Heaviside function in the standard way $H(x)=\begin{cases} \ 1 & x\geq 0 \\ \ 0& x<0 \end{cases}$ Then I want to find the Wavefront set where I am using the definition ...