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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18 views

Why are the following limits used?

exercise in general says Find k if the joint probability density of X, Y, and Z is given by $$f(x,y,z)=kxy(1−z) \text{ for } 0<x<1, 0<y<1, x+y+z<1$$ then the answers define the ...
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73 views

A problem with Lebesgue integral in functional analysis

Let $ F \in D '$ have a compact support, $ F(\varphi)\geq0$ for any $\varphi \geq0$. How can i prove that $F(\varphi) = \int \! \varphi \, \mathrm{d}\mu$ for some non-negative measure $ \mu $?
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19 views

Does the extension of an element of $W^k$ by $0$ still lie in $W^k$?

Let $U\subset\mathbb{R}^n$ be an open set and $Z$ a closed subset of $U$ . We denote by $W^{k}(U)$ the Sobolev space of functions whose derivatives (in the sense of distribution theory) up to order $k$...
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18 views

How do I solve the distributional equation T.x =1?

I am having a bit of trouble solving distributional equations. An example that I am currently working on is to show that the distributional equation $T.x = 1$ has a solution if and only if $T=p.v.(\...
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1answer
16 views

Show that $\partial_j f$ exists at any point and is equal to $\partial^{dist}_{j} f$

Show that f $f\in C(\mathbb{R}^n)$ and $\partial^{dist}_{j} f\in C(\mathbb{R}^n)$ then $\partial_j f$ exists at any point and is equal to $\partial^{dist}_{j} f$. Where $\partial^{dist}_{j}$ is the ...
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29 views

An $L_1$-bounded, divergent sequence of divergence-free vector fields

I am looking for examples of $L_1$-bounded sequences of vector fields in $L_1=L_1(\mathbb R^n, \mathbb R^n)$ ($n>1$) that have zero divergence (in the distributional sense) such that no subsequence ...
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1answer
48 views

A few questions about topologies on $\mathcal{C}^\infty_0 (\Omega)$

Reading about the space $\mathcal{C}^\infty_0(\Omega)$ of all compactly supported functions, I've came across a claim that this space is not complete with respect to the family of seminorms $$ \|\...
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1answer
36 views

Proving that a regular distribution is in fact a distribution.

Given $\Omega \subset \mathbb{R}^{N}$ and a locally integrable function $f$, its regular distribution $T$ is defined by $\langle T,\phi \rangle = \int_{\Omega}f(x)\phi(x)dx$, where $\phi \in D(\Omega)$...
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30 views

Prove that a given distribution is tempered

I have a distribution $E$ such that $\phi \ast E$ is tempered for all $\phi \in C_c^\infty$. Is it possible to prove that $E$ is tempered? Something along the lines \begin{equation} \lim\limits_{n \...
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22 views

When is the product of a Schwartz function and a smooth function guaranteed to be a Schwartz function?

How can we classify functions $g \in C^\infty \left(\mathbb{R}^d\right)$ such that for all Schwartz functions $f$ on $\mathbb{R}^d$, $f g$ is again Schwartz? I believe this question could be answered ...
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1answer
41 views

Finding the distribution which solves a given equation [closed]

I know how to find a $u$ in the space of distributions on $\mathbb{R},$ i.e. $D'(\mathbb{R})$ which solves $xu=1.$ But I can't seem to be able to find $u$ in $D'(\mathbb{R})$ which solves $(x^2)u=1.$ ...
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4answers
74 views

Limit $\lim_{\epsilon \rightarrow0} \frac{1}{\sqrt{\epsilon}}\exp\left({-\frac{x^2}{\epsilon}}\right)$ in the sense of distributions [closed]

Compute the following limit: $$\lim_{\epsilon \rightarrow0} \frac{1}{\sqrt{\epsilon}}\exp\left({-\frac{x^2}{\epsilon}}\right)$$ I do not know how the "sense of distributions" is applied in this ...
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1answer
119 views

Wave equation and Fourier Transform: conditions for differentiation

Consider the wave equation in one dimension $u_{tt}-u_{xx}=0$ together with a Fourier Transform along $t$, ie $$\text{FT}[u](x,\omega)=\int_{-\infty}^{+\infty}u(x,t)\exp(-i\omega t)\mathrm{d}t.\tag{1}$...
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3answers
90 views

Limit in distributions of $\frac{\sin(tx)}{x}$

How can I find the limit of $\frac{\sin(tx)}{x}$ as $t \to \infty$ in $D'$ ? I understand that i need to see the $\lim_{t \to \infty}{\int_{\infty}^{\infty}{\frac{\sin(tx)\phi(x)}{x}dx}}$ for every ...
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0answers
19 views

An extension problem in Sobolev spaces

Suppose $u\in H^{-s}(0,1)$ and $0<s<1/2$. Is there $\tilde{u}\in H^{-s}(\mathbb{R})$ such that the restriction of $\tilde{u}$ to $(0,1)$ is $u$? Does this work: $\tilde{u}(\phi)=u(\phi|_{(0,1)})...
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57 views

Convergence of sequences in D' is not metrizable

I'm struggled with the problem. I need to proove that the convergence of the sequence of generalized functions in D' can't be defined by any metric (can't be metrizable). I've tried to use the fact ...
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1answer
29 views

Tempered distributions and linear transformations

I am reading in my lecture notes about Tempered distributions and one of the statements goes as follows: Let $T$ be a Tempered distributions, define the shifted distribution $T_{\{a, L\}}(f)=|\...
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1answer
20 views

Distributional equation $p(x) T = \varphi(x)$, $p$ polynomial, $\varphi$ smooth with rapid decrease

I am posing a variant of this question. Data: a polynomial $p(x)$, $x\in\mathbb{R}$, with complex coefficients, and function $\varphi:\mathbb{R}\to\mathbb{C}$, smooth, with rapid decrease (say, ...
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17 views

Show the singular support of $\delta(p(x))$ agree with it's support

Let $S=\{x\in \mathbb{R}^n| p(x)=0, \nabla p(x)\not= 0, p(x)\in C^{\infty}\}$. We defined the distribution $\delta(p(x))\in \mathcal{D}'(\mathbb{R}^n)$ on $S$ as follows: for all $\varphi\in C_{0}^{\...
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2answers
48 views

Why $\pi$ appears in Dirac? [duplicate]

"With hands", what is the reason why $\pi$ appears in this result : $\lim_{\epsilon\to 0}\frac{\epsilon}{\epsilon^2+x^2}=\pi\delta(x)$
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26 views

Derivative of $\delta_{f(x)}$ distribution

I asked this question. Basically I want to calculate the derivative of $\delta_{f(x)}$ with respect to $x$, that is $D=\left \langle \dfrac{d}{dx}\delta_{f(x)} ,\varphi\right \rangle $. In the answer ...
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1answer
59 views

Derivative of a delta function reference

This question has been asked many times. I've read that $\dfrac{d(\delta(f(x)))}{dx}=-f^{'}(x)$. For instance, here, the second answer. I am coming from the context of probability. In particular, I am ...
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1answer
35 views

What is the best current summary of the limits of (tempered) distribution theory where it comes to multiplication of distributions?

As I understand it, it has been shown that a consistent calculus of (tempered or otherwise) distributions involving multiplication cannot be constructed. Since the problem seems central to the theory ...
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1answer
77 views

Limit of distribution functions

Let $D$ be a space of distribution functions. I need to prove that for any function $F \in D'$ there are distribution functions $F_n \in D$ for which $F_n(f) \rightarrow F(f)$ for every $f \in D$. I ...
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1answer
52 views

Approximation of a distribution from $\mathcal{D}'$ by functions from $\mathcal{D}$

I need to prove that for any generalized function ( they are also called distributions) $f \in \mathcal{D}'$ there exists generalized functions $f_n \in \mathcal{D} $, given by normal functions from $...
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1answer
44 views

Derivative of a distribution in $\mathcal{D}'(\mathbb{R})$

How do I show that the Newton's quotient is still valid for distribution? By this, I mean that if $u \in \mathcal{D}'(\mathbb{R})$ and if $u_r$ is the translation $\langle u_r,\phi \rangle = \langle u,...
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1answer
47 views

The Dirac measure as a weak limit of $L^2$ functions on a LCA group.

Let $K$ be a locally compact abelian group. In the proof of Proposition 2 (the proposition does not matter for my question) of this blog-post, Tao writes: $K$ comes with an invariant probability ...
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1answer
45 views

Fourier transform of the dilation of a distribution

Let $F$ a tempered distribution, in my textbook the transform of $F$ is defined by $\hat{f}[ \phi]=F[\hat{\phi}]$ , having said that l'm interested in prove the following property of the Fourier ...
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77 views

John Horváth - Topological Vector Spaces and Distributions Volume 2

In at least three places in the book Topological Vector Spaces and Distributions, written by John Horváth, the author comments on a second volume of his book that has never been published. In ...
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1answer
29 views

when defining continuous operators on distributions, why does the adjoint operator need to be continuous?

Here is how folland introduces the construction of continuous operators on distributions: I understand that the adjoint/equality thing is needed so that the constructed operator is an extension of ...
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0answers
42 views

Discontinuity multiplied by a distribution. Does the sum of undefined pieces yield a defined one?

Henceforth, let $\delta$ be the Dirac delta and $\theta$ the Heaviside. While the multiplication, in the distributional sense, $\langle\psi\delta,f\rangle$ with smooth $\psi \in C^{\infty}(\mathbb{R}...
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1answer
21 views

Proof that a zero-variance Gaussian function becomes a Delta distribution

Consider the Gaussian function: $$ f_{N(\mu, \sigma^2)}(t) = \frac{1}{\sigma \sqrt{\pi}} \exp \left[ {-\frac{1}{2}\left( \frac{t - \mu}{\sigma} \right)^2} \right] $$ I have seen in some texts ...
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1answer
30 views

Cartesian product of Fréchet

Let $F_{1},...,F_{n}$ be Fréchet spaces. Is the cartesian product $F_{1}\times\cdots \times F_{n}$ again a Fréchet space? In particular, is the cartesian product $\mathcal{S}(\mathbb{R}^{d})\times\...
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1answer
20 views

Radial Schwarz functions are dense in radial tempered distributions

I am trying to show that any radial tempered distribution can be approximated by radial Schwarz functions, where $T\in{S}'(\mathbb{R}^n)$ being radial means $\langle{T},\phi\circ{R^T}\rangle=\langle{T,...
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1answer
35 views

What is the asymptotic form of the spherical Bessel function $j_{n}(x)$ when $n \to \infty$?

I am trying to find the asymptotics of the spherical Bessel function $j_n(x)$ when $n\to \infty$. I was able to find something like $$ j_{n} ( x) \sim \sqrt{\frac{\pi}{2n}} \delta \left(x - n\...
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0answers
30 views

Understanding the trace operator and distributional product

I'm working now on weak formulation of a linear elliptic pde and I want to know what is the difference between: i) Trace operator $\gamma_{0}$ and restriction of a function ii) Distributional ...
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40 views

How I can calculate this integral

The integral is the following and I want to produce it in terms of $r$. Is there a theory that the partial producer enters the integral? $$\frac{\partial }{\partial r} \int_{F^{-1}(p)}^{\infty} \...
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2answers
103 views

Solving the “Transport” PDE in the sense of distributions with Dirac Delta Source

Let $\delta_0$ be the standard Dirac Delta distribution. I wish to solve the PDE $$u_t+cu_x=\delta_0$$ in the sense of distributions with initial condition $u(x,0)=g(x)$ for some continuous $g$. That ...
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1answer
16 views

Partial Derivatives of a continuous function is the Dirac distribution [closed]

I am having trouble with the following question and a detailed solution would be immensely appreciated: Determine a continuous function $f:\to \mathbb{R}^n \to \mathbb{F}$ and a multi-index $ \alpha ...
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0answers
22 views

Vanishing Fourier transform and bounded functions

Let us consider $f$ a function in $C(\mathbb{R})\cap L^{\infty}$ admitting a Fourier transform (say, either in the $L^1$ sense or tempered distribution sense if $f \notin L^1$). We assume that $supp(\...
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1answer
25 views

Making sense of covariance of space-time white noise as a product of delta distributions

The covariance of space-time white noise $\dot{W}(x,t)$ is given by $\mathbb{E}\dot{W}(x,t)\dot{W}(y,s) = \delta(t-s)\delta(x-y)$, where the $\delta$-distribution satisfies $\delta(x) = 0$ if $x\neq 0$...
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1answer
36 views

What is the formal definition of the singular support of a distribution?

The definition I have is that:For a distribution $u \in \mathcal{D}'(U)$ where $U$ is an open subset of $\mathbb R^n$, a point $x$ is in the singular support of $u$ if $u$ is not smooth on an open set ...
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0answers
30 views

Show that $\langle T, \varphi \rangle$ is a tempered distribution

Show that $\langle T, \varphi \rangle = \sum_{n=1}^{\infty} \varphi(n)$ is a tempered distribution on $\mathbf{R}^1$. My question is from Strichartz book, A guide to Distribution theory and Fourier ...
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0answers
10 views

Distribution theory and Shannon sampling theorem

Let $C$ denote the Dirac comb distribution and let $\mathcal{F}$ denote the Fourier transform for tempered distributions. Let $x$ be any function in the Schwartz class $\mathcal{S}$ with $X = \mathcal{...
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1answer
20 views

Convolution of compactly supported distributions

Is the convolution of two compactly supported distributions of non disjoint supports always defined? If yes is it itself compactly supported?
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1answer
24 views

When is the distributional derivative a function?

Let $u \in D'(\mathbb{R})$ be a distribution and suppose that its distributional derivative $u'$ can be identified with an $L^1_{\mathrm{loc}}$ function. Can the distribution $u$ itself then also be ...
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1answer
23 views

Dirac distribution and Sobolev spaces

I see the conclusion that $\delta_{x_{0}} \in H^{s}\left(\mathbb{R}^{n}\right)$ if and only if $s<-n/2$, where $ H^{s}\left(\mathbb{R}^{n}\right) $ is Sobolev space. from many places, and the ...
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0answers
19 views

Reference for some facts relating on functional analysis on manifolds

I am looking for references where I can find proofs for the following statements. Also, if somebody could point out how these statements relate to each other, I would be very grateful. Let $M$ be a ...
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0answers
20 views

Convergence of smoothing operators, Schwarz seminorms and mistakes in the book Modern Fourier Analysis

The question is about exercise 8.3.2 part (c) of Grafakos's book Moder Fourier Analysis. The hint seems to be wrong and as there has been a couple of situations like this with the book I want to make ...
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1answer
44 views

uniform convergence of a Fourier series to a continuous periodic function on $[-\pi,\pi]$

with the help of Fourier series theory on $[-\pi,\pi]$, show that for any given periodic continuous function (complex valued) $f$ on $[-\pi,\pi]$ of period $2\pi$ and $\epsilon>0$, there exists $P=\...

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