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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21 views

2nd order distribution differential equation

Denote by $f(x)$ a quadratic polynomial with coeffitients $a, b, c$: $$ f(x) = ax^2 + bx+ c$$ And let's look at distribution differential equation $$f \left(\dfrac{d}{dx}\right) y =\text{p.v.} \dfrac{...
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Principal symbol of the conjugate of a partial differential operator

Let $P=\sum_{\alpha}p_{\alpha}(x)D_x^{\alpha}$ be a partial differential opperator on $\mathbb{R}^m$ where ${\alpha}\equiv(\alpha_1,...,\alpha_m)$ is a multi-index and $D_x^{\alpha}=(-i)^{|\alpha|}\...
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45 views

Grafakos, Modern Fourier Analysis, Third Edition, Exercise 2.1.4. Bounded Tempered Distributions

In Grafakos book, Modern Fourier Analysis, the exercise 2.1.4 is as follows: Let $P$ be the Poisson Kernel. Show that for any bounded tempered distribution $f$ we have $P_t \ast f \to f$ in $\mathcal{...
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54 views

Convolution between a Bounded Tempered Distribution and a $L^1$ function is a distribution.

In this question we can see the definition of a convolution between a bounded tempered distribution and a $L^1$ function. In the comments, there are two possible ways to show that this is in fact a ...
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1answer
22 views

Lebesgue Integral of slowly incrasing polynomial on $\mathbb{R}^m$

I found that this fact is used everywhere in the study of Sobolov spaces and distribution theory: $\frac{1}{(1+|x|)^N}$ is integrable on $\mathbb{R}^m$ if $N>m$. But I am unable to find a proof for ...
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23 views

Canonical LF topology

Wikipedia: The space of test functions usually consists of smooth functions with compact support that are defined on some given non-empty open subset ${\displaystyle U\subseteq \mathbb {R} ^{n}}$. ...
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10 views

On analytic test functions

In what follows, ${\cal C}^\infty_c(U)$ is intended to be the set of all smooth functions with compact support defined on a non-empty open subset $U \subseteq \mathbb R^n$. Wikipedia: There $[\dots]$ ...
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34 views

Is it true that $\widehat{f(x,0)}(\xi) = \widehat{f}(\xi,0)$?

Let $f \in S(\mathbb{R}^2)$ and $g(x)= f(x,0)$. What is the relationship between $\widehat{g}$ and $\widehat{f}$? It's clear that $g \in S(\mathbb{R})$ so both $f$ and $g$ have a Fourier transform. I ...
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25 views

The convolution of two distributions

I'm looking to find the Fourier transform of the distribution $$|x|^{-n+q+2}|Tx|^{-1}$$ where $-1 < q < n-2$, $T$ is a linear transformation, and $|\cdot|$ is the Euclidian norm on $\mathbb{R}^n$...
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1answer
30 views

For a function $f = f(t,x)$, how is the space $C^0([0,T];(S'(\mathbb R^d))^n)$ defined?

I'm reading "Fourier Analysis and non-linear Partial Differential Equations" from H. Bahouri and they use a notation which is unknown to me. If $(t,x) \in \mathbb [0,T] \times \mathbb R^d$, $...
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Compute $\int dq \ q^{n} j_{\ell} \left( q r \right) j_{\ell}\left( q R \right)$

I am interested in analytical formulas for integrals of products of spherical Bessel functions times a power: $$I_{n,\ell}(r,R) \equiv \int_0^{\infty} dq \ q^{n} j_{\ell} \left( q r \right) j_{\ell}...
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1answer
88 views

Sokhotski–Plemelj theorem for two-dimensional integrals

I am doing some physics calculations, and I need to deal with integrals that can be reduced to this general form: $$ \lim_{\epsilon \to 0^{+}} \int_{-a}^{b} \int_{-a}^{b} \frac{f(x,y)}{(x-i\epsilon)(y-...
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Deduce if $\dfrac{dT}{dx} \in \mathcal{E}(\mathbb{R})$ for $T \in \mathcal{D}'(\mathbb{R})$ then $T \in \mathcal{E}(\mathbb{R})$.

Deduce if $\dfrac{dT}{dx} \in \mathcal{E}(\mathbb{R})$ for $T \in \mathcal{D}'(\mathbb{R})$ then $T \in \mathcal{E}(\mathbb{R})$. My attempt : We know for any $T \in \mathcal{D}'(\mathbb{R})$ , if $T'...
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Distributional Limit $T_{g_{n}} \rightarrow \delta_{0}$

Consider $$ \psi(x)=\left\{\begin{array}{ll} e^{-\frac{1}{1-x^{2}}}, & |x|<1 \\ 0, & |x| \geqslant 1 \end{array}\right. $$ and $$ g_{n}(x)=\frac{\psi(n x)}{\displaystyle\int_{-\infty}^{\...
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proof of the product: x times m derivative of delta function.

The question is in one dimension and is : Prove that $$x\delta^{(m)}=-m\delta^{(m-1)},\ m \in \mathbb{N},$$ where $\delta^{(m)}$ is the $m$-derivative of $\delta.$ As I know, I got through this way: $...
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10 views

Remainder term of Taylor expansion in distribution theory

I want to regularise an integral in the sense of distributions, given by $$I= \int_{x}^{\infty} dy \, (y-x)^{\lambda} p(y) \varphi \left(\frac{y}{q}\right) \;, $$ where $\lambda \in \mathbb{R}$, so ...
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15 views

Generate correlated array using given array and correlated coefficient

Given array, $A = [1,2,3,4]$. Let $A' = [3,2,1,4]$. Then the Pearson correlation coefficient is $(A,A') = 0.2$. So can we generate $A'$ using correlation coefficient $(0.2)$ and $A([1,2,3,4])$?
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60 views

Is there a metrizable topology on tempered distributions for which sequential convergence is pointwise?

Is there a metrizable topology on $S’$ (the set of linear continuous functionals in Schwarz’s space) whose convergent sequences are the sequences that converge pointwise? The obvious topology with ...
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70 views

Sufficient conditions for derviative of this product to be in $L^2$.

Let $I=(l_0,l_1)$, $y \in H^2(I)$ and $\rho(x) \in L^\infty(I)$ with $\rho(x)>c\geq 0$. Define $D_\rho(x)=\rho(x)\chi_{(a_0,b_0)}$ with $l_0\leq a_0<b_0\leq l_1$. Now, For any $\phi \in C_c^\...
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62 views

How does derivative transfer to test function?

I set out to find a fundamental solution $E$ for the Poisson equation, i.e. a distribution $\mathscr D'(\Bbb R^d)$ such that $\Delta E = \delta$; I'm almost done. The only thing I have left to do is ...
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1answer
89 views

Show that $f(x) = e^x \cos(x)$ on $\mathbb{R}$ is a tempered distribution

As shown in the title. I know that the anti-derivative of $f(x) = e^x \cos(x)$ is $ \frac{\sin(x)+\cos(x)}{2} e^x$, whose anti-derivative is $\frac{\sin(x)}{2}e^x$...but not sure how to prove it.
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On the definition of the Radon measure $-\Delta u$ when $u \in H_0^1, - \Delta u \geq 0$

Let $D \subset \mathbb R^N$ be an open set and $u \in H_0^1(D)$ such that $- \Delta u \geq 0$ (as a distribution). Then $- \Delta u$ defines a Radon measure. If $w \in C_c^\infty(D)$, one can write $$ ...
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38 views

Find the Green's Function for $x^2y''(x)+xy'(x)-y(x)=f(x)$ for $0<x<1$ with $y(0)=0$ and $y(1)=1$

Let $G(x;\lambda)$ be the Green's function. We require that $x^2G''(x;\lambda)+xG'(x;\lambda)-G(x)=\begin{cases} \mbox{$0$} & \mbox{if } x\in(0,\lambda) \\ \mbox{0} & \mbox{if $x\in (\lambda,1)...
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16 views

Approximation of generalized function

For every generalized function $F$ exist sequence of functions $F_{k} \in \mathcal D$ such as for every $\phi \in \mathcal D$ true that $F(\phi)= \lim\limits_{k\to\infty} F_{k}(\phi)$ Is that true and ...
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1answer
66 views

Scaled Dirac Delta function: $ \delta (xe^r - y) $

I was reading on squeezed Gaussian states and stumbled upon this paper: Equivalence Classes of Minimum-Uncertainty Packets. II. It is mentioned after Eq. $\left(2\right)$ that $$ \left\langle x\left\...
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1answer
68 views

find the limit in distributions(space of generalized functions)

How to find limit in $ \mathcal{D}^{'}$ $$ \exp(itx)(x+i0)^{-1}, t \rightarrow \infty $$ I try to use Sokhotsky's formula $(x+i0)^{-1} = -i\pi\delta(x) + \rho\frac{1}{x} $ , but did not come to a ...
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1answer
54 views

Is it correct to set $\delta(t) f(t) = \delta(t) f(0)$ within a distribution?

I have some complicated probability distributions which come out as $$ P(x,t) = \delta(t)G(x,t) + K(x,t),$$ where $G(x,t)$ and $K(x,t)$ are continuous in time. Is it permissible to simplify such ...
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36 views

Poincaré Inequality on Gaussian Measures

So I have a working idea on Gaussian-Poincaré Inequality. Namely through the Ornstein-Ullenbeck Generator and Gaussian Integration by parts. Recently I have stumbled across Sobolev Spaces and have ...
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16 views

Which distribution kernels represent continuous linear operators $\mathcal{D}(X)\to\mathcal{D}(Y)$?

If $X$ and $Y$ are smooth manifolds, which distribution kernels $K$ represent continuous linear operators $\mathcal{D}(X)\to\mathcal{D}(Y)$? According to Theorem VII in Schwartz's 1950 ICM paper, &...
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1answer
38 views

Distributions dual to functions of polynomial growth.

The space of distributions on $\mathbb{R}^n$ is essentially found by requiring that it should be possible to apply the distribution to any bump function. Similarly, compactly supported distributions ...
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28 views

$-\int_a^b F_{\mu}(x)\phi'(x)dx=\ldots=\int_a^b\phi(x)d\mu(x)$?

If I consider the interval $[a,b]$ and a positive measure $\mu$ on this interval, we can define a function $F_{\mu}(x)=\mu([a,x])$. I want to show that the distributional derivative of this function ...
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50 views

Determine $p$ such that $u(x,y)=1-\max\{|x|,|y|\}\in W^{1,p}((-1,1)\times(-1,1))$

Let $\Omega$ be the open square $(-1,1)^2\subset\mathbb{R}^2$, $u\in L^1(\Omega)$ such that for each $(x,y)\in\Omega$, $u(x,y)=1-\max\{|x|,|y|\}$. Determine the weak gradient of $u$ and find $p$ such ...
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1answer
44 views

Determine $a$ and $p$ such that $u(x)=|x|^{-a}\in W^{1,p}(B_1(0))$

Let $\Omega=B_1(0)=\{x\in\mathbb{R}^N:|x|<1\}$ and let $u\in L^1(\Omega)$ such that $u(x)=|x|^{-a}$, with $0<a<N$. Determine $\nabla u$ as distributional derivative. Then determine $a$ and $p$...
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1answer
122 views

Zero distributions

A distribution, by the definition given in Shearer and Levy, is a function $f \in D'(\mathbb{R})$ with $f: C^{\infty}_c(\mathbb{R})=D(\mathbb{R}) \rightarrow \mathbb{R}$ s.t. $f$ is linear $f$ is ...
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20 views

Product rule of distribution by multiplicating a $C^\infty$ function

I am reading S. Kesavan's book "Topics In Functional Analysis and Application" and trying to prove the product rule, can you check where I am going wrong? Let $\psi \in C^\infty (\mathbb{R})$...
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1answer
75 views

Function which integrates to $0$ against test functions with mean $0$ is constant almost everywhere.

Suppose $U$ is a bounded domain in $\mathbb{R}^n$ and $u \in L^1(U)$ has the property that $$\int_{U}u\phi=0 $$ for all $\phi \in C_{c}^{\infty}(U)$ which satisfy $\int \phi = 0$. I'd like to show ...
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40 views

Prove $u, \Delta u \in L^2(\mathbb{R}^n)$ then $u \in H^2(\mathbb{R}^n)$

Let $k=2$ throughout. Let $u, \Delta u \in L^2(\mathbb{R}^n)$ then $u \in H^k(\mathbb{R}^n)$ where $H^k= \{u \in S'(\mathbb{R}^n): (1+|\xi|^2)^{\frac{k}{2}} \widehat u \in L^2\}$. I am a bit unsure ...
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2answers
64 views

Distributional derivative of multiple step function

i am learning about distributions and distributional derivatives, and would like to check my understanding. suppose we have a step function that jumps by a unit at every integer: $$H(x)= \begin{cases} ...
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39 views

Proving $\delta \notin C_c$

Prove there is no $\delta \in C_c(\mathbb{R})$ such that for all $f \in C_c(\mathbb{R}) \ f= \delta \ast f$. I think I have a solution to this problem, but am a bit unsure about it, as the author of ...
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1answer
25 views

Distributional derivatives of euclidean norm

does anybody know how to calculate the distributional derivatives of the function: $f: (x_1,x_2,...,x_n)\to \sqrt{x_1^2+...+x_n^2}$? I only know the way for $n=1$, but I don't see how to do this in ...
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40 views

Continuity equation, distributional derivative = usual derivative?

We say that a family of pairs measures/vector fields $(\rho_t,v_t)$ with $v_t\in L^1(\rho_t;\mathbb{R}^d)$ and $\int_0^T\|v_t\|_{L^1(\rho_t)}dt=\int_0^T\int_{\Omega}|v_t|d\rho_tdt<+\infty$ solves ...
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23 views

Does convergence of distributions to $0$ in $D(\mathbb R^d \setminus \{0\})$ implies convergence in $D'(\mathbb R^d)$?

Let $(u_n)_{n=1}^{\infty} \subset D'(\mathbb R^d)$ (or $\subset S'(\mathbb R^d)$) be a sequence of distributions. Assume that the sequence $(\left. u_n \right|_{\mathbb R^d \setminus \{0\}}) \subset D'...
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19 views

Delta function limit definition in higher dimensions

I know the following definition of the $\delta$-function: Given $x\in\mathbb{R}$ we have \begin{equation}\label{eq1} \delta(x)=\frac{1}{2}\lim_{\epsilon\rightarrow 0}\epsilon|x|^{\epsilon-1}. \end{...
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1answer
57 views

Is there a 'distributional antiderivative'?

Suppose I have a distribution $F$ which I know on the derivative of an arbitrary test function. That is, I know $F(\phi')$ for all $\phi ∈ \mathcal{D}(ℝ)$. I also know that $F(\phi') = -F'(\phi)$. I ...
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20 views

Weak convergence of measures on the unit sphere $S^2$.

Let $(x_i,y_i) \in S^{\mathbb{N}} \times S^\mathbb{N}$ where $S$ is the unit sphere in $\mathbb{R}^3$. Assume the following result of weak convergence for measures : $$\frac{1}{N} \sum_{i=1}^N \delta_{...
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81 views

Delta function singularity

I am reading the Wiki article about singularities and I was wondering what kind of singularity is the Dirac delta function not defined as a distribution but as this way: \begin{equation} \delta^{\...
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113 views

$-\Delta u+\epsilon \sin u=f$ has a unique solution in the sense of distribution

It's the last question in my exam. Prove that there is a neighborhood of $0$, denoted by $[-\delta,\delta]$, $\forall \epsilon \in [-\delta,\delta]$ the following equation has a unique solution in ...
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1answer
79 views

Show that if $g \in \mathcal{D'}(\mathbb{R})$ satisfies $\frac{dg}{dx}=0$ then g is constant.

I'm having trouble figuring this out and its been bothering me for a few weeks now. The solution constructs a test function in a particular way to prove the result, but I have trouble understanding ...
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1answer
22 views

Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, $F(t)=\int_{x_0}^{t}f(s)ds$. Prove that, $DF=f$ (towards theoretical distribution).

Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, consider $$ F(t)=\int_{x_0}^{t}f(s)ds. $$ Prove that, $DF=f$ (towards theoretical distribution). I thought of the following: Let $\...
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0answers
39 views

How to define the grad, div and the curl of a distribution?

I have learned how to define the derivative or partial derivative of a distribution, but I still can't find a clear definition of the grad, div and curl of a distribution, I would appreciate it if you ...

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