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
113 views

I want to prove $f\notin W^{1,1}(\mathbb{R},\gamma_{1})$

Let $\gamma_{1}=\mathscr{N}(0,I_{1})$ in $\mathbb{R}$ be the standard Gaussian measure. Consider the sequence $(f_{n})_{n\in\mathbb{N}}\in C_{b}^{1}(\mathbb{R})$ defined by $$f_{n}(x)=\begin{cases} ...
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1answer
62 views

Does this function define a distribution? $f(x) = \sum_{n \in \mathbb{N}} e^{ -(x-n)^2 }$

Is this a continuous linear form on $\mathcal{D}(\mathbb{R})$ ($C^\infty$ with compact support) such that for any sequence $\varphi_n\to0 \Rightarrow \langle T, \varphi_n \rangle \to 0$. ...
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1answer
40 views

Prove that a distribution has its primitive distribution.

Given a distribution $F$ in $\mathbb{R}$,prove that there exists a distribution $F_1$ such that $$\frac{d}{dx}F_1=F$$,and it is unique up to an additive constant.
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0answers
30 views

Conditionally convergent distributions?

The notion of conditional convergence can be extended to integrals. Can it also be extended to distributions - specifically for tempered distributions? The motivation behind this question comes from ...
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0answers
15 views

Fourier transforms of some homogeneous functions

In $2D$, what is the Fourier transform (in the sense of distributions) of functions of the form $x_i/|x|^2, 1/|x|, x_i/|x|, x_i x_j /|x|^2$, and so on? Here, $i = 1,2$. They are homogeneous and ...
2
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1answer
35 views

Convergence of complex exponential in dual Schwartz space

I am trying to solve the following problem, but I am a little bit missed. Show that the functions $e^{inx}$ and $e^{-inx}$ converge to zero in S' (dual Schwartz space) as $n → ∞$. Conclude that ...
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0answers
21 views

Representation of functional/distribution through infinte series of delta distributions possible?

Sorry, I am an engineer and this surpasses completely my math education. If I am not precise, please comment on open/unclear assumptions, and I will do my best in correcting my mistakes. QUESTION: is ...
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2answers
63 views

Support of a distribution, what does it mean?

In my course notes the support of a distribution (continous lineair functional) is defined as follows: Definitions First it defines something like open annihilation sets: An open annihilation ...
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1answer
61 views

First and second derivatives of max function

I have two functions $f(x)=(x-K)_+$ and $g(x) = \max\{x,K\}, x \geq 0, K = const \geq 0$. I was told that $$f'(x) = \mathbb{I}_{[K,+\infty)}(x)$$ and $$f''(x)=\delta_K(x),$$ because ...
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0answers
36 views

Fourier transform of distribution solution

Let $f(x)=2$ for all $x$. What is the Fourier transform of $f$? This is my solution but there are some steps I don't fully understand, I took it from an example just to get through the rest of the ...
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2answers
43 views

Radial distributions

here is a theorem I am not able to solve, it is about distributions (Schwartz). Let $\Omega = \mathbb{R}^2 \setminus \{0\}$. Show that for all $S \in \mathcal{D}'(\mathbb{R}_+^*)$ there exists a ...
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1answer
38 views

A simple calculation about distribution in the plane $\mathbb{R}^2$.

Let $\Omega = \lbrace (t,x)\in\mathbb{R}^2\mid t>|x| \rbrace$ and $T\in\mathcal{D}'(\mathbb{R}^2)$ defined by $T=\partial_t\mathbb{1}_\Omega - \partial_x\mathbb{1}_\Omega$ Prove that ...
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0answers
48 views

Proving an identity of the composition of the delta distribution with a differentiable function

Given a differentiable function $f$, some $x_j$ ($j \in \{1, ..., n\}$) such that $f(x_j) = 0$ $\forall j$ and $f'(x_j) \ne 0$ $\forall j$, and the following definition of the composition of a ...
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1answer
43 views

reference for periodic distributions $\mathcal{D}(\mathbb T).$

I am looking for some introduction to theory of periodic distributions $\mathcal{D}(\mathbb T).$ Would you please suggest some reference book? [ I am familiar with distibutions on $\mathbb R$. ...
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0answers
34 views

Prove that a homogeneous distribution is tempered.

Suppose $F$ is a homogeneous distribution of degree $λ$.Prove that $F$ is tempered,i.e.$F$ is continuous in the Schwartz space $\mathcal{S}$. It seems that it's an easy result in distribution ...
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0answers
28 views

Distributional derivative $Xu=0$ then $u$ is constant over the flow lines of $X$

Suppose we have a Lie group on $\mathbb R^n$, something like $(\mathbb R^n, \cdot)$, where with $\cdot$ we denote the group law on $\mathbb R^n$. Call $\mathfrak g$ the Lie algebra of $(\mathbb R^n, ...
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0answers
28 views

A sufficient and necessary condition for a distribution to be tempered.

Show that the distribution $F$ is tempered if and only if there is an integer $N$ and a constant $A$,so that for all $R\geq 1$, $$F(\varphi)\leq AR^N sup_{|x|\leq R,0\leq |\alpha|\leq ...
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1answer
58 views

Are the three statements the same?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz function on $\mathbb{R}^n$. Consider two statements which have the same proof. $$ f\in \mathcal{S}(\mathbb{R}^n)\,\,\Longrightarrow\,\,f\in ...
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1answer
91 views

The Schwartz function and the sobolev space $W^{2,p}$

How do you prove the Schwartz functions in $\mathbb{R}^n$ are dense in the space $W^{2,p}(\mathbb{R}^n)?$ Terrence tao has a version of the proof of The space $C_c^{\infty}(\mathbb{R}^d)$ of test ...
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2answers
82 views

If $f,g_j\in\mathcal{C}(U)$ and $\frac{\partial f}{\partial x_j}=g_j$ weakly $\Rightarrow$ $f\in\mathcal{C}^1(U)$

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f,g\in \mathcal{C}(U)$. If $$\frac{\partial f}{\partial x_j}=g$$ for some $j$ $(j=1,\ldots,n)$ in the sense of distributions, how to prove that ...
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0answers
42 views

an iff proof on the existence of weak derivative

I have trouble understanding the following proposition. Proposition $f,g\in L_{\text{loc}}^1(\Omega)$. Then $g=D^{\alpha}f$ iff. there exists $f_m\in C^{\infty}(\Omega)$ such that $f_m\to f$ in ...
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1answer
49 views

Delta function in the sense of distributions

I have a problem understanding the meaning of the delta-function on the sense of distributions. E.g. I have the following equation: $$\left(\frac{d}{dt} \theta(t) \right) f(t) = \delta(t) f(t)$$ ...
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1answer
67 views

Fourier transform of the identity function $f(x)=x$

Let's say you are given $\omega_f \in \mathcal{S}'(\mathbb R)$ with \begin{align*} f \colon \mathbb{R} &\to \mathbb{R}\\ x &\mapsto x, \end{align*} and the definition of Fourier transform ...
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1answer
13 views

How to determine a function from its corresponding distribution?

If we have a function $\phi(x)$ we can determine the corresponding distribution $\phi^D$ such that: $$\forall f:L_{\phi^D}(f)=\langle\phi^D|f\rangle=\int_\mathbb{R}\phi(x) f(x) dx$$ as long as $f$ ...
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0answers
52 views

Solutions of an EDO in tempered distribution space being smooth out of the origin.

For $a\in\mathbb{C}$, let us consider the following differential equation in $\mathcal{S}'(\mathbb{R})$, the set of tempered distributions on $\mathbb{R}$: $$xT''+2T'+(a-x)T=0.$$ For ...
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1answer
30 views

Fourier Transform of the “regular” tempered distribution of $|x|$

As the title states I am trying (without luck) to compute the Fourier Transform (in tempered distributional sense) of $|x|$, meaning ($\mathcal{S}(\mathbb{R})$ the Schwartz space): ...
1
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1answer
48 views

Inverse convolution of a distribution.

Notation. Let ${\mathcal{D}'}_+(\mathbb{R})$ be the set of distributions on $\mathbb{R}$ supported on $[0,+\infty[$. One easily derives the: Proposition. Let ...
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0answers
29 views

Variational formulation curl-div equation

I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ ...
2
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1answer
63 views

Is $f(x)=e^x \cdot \cos(e^x)$ a tempered distribution?

Let $f(x)=e^x \cdot \cos(e^x)$. Define $$T_f(\varphi)=\int_{-\infty}^{+\infty} f(x) \cdot \varphi(x) \ .$$ I would like to know if $T_f$ defined with the formula above defines a tempered distribution ...
2
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1answer
148 views

Placing delta's at maxima, Is there any smart equation based expression?

Let $M$ be the set of maxima of the function $k:\mathbb{R}\to \mathbb{R}$. We define the function $$L(t) = \sum_{y\in M} k(y)\delta(t-y)$$ and there by the step function $$\Gamma(t) = \int_0^t ...
1
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1answer
72 views

Tempered distributions and convergence

It is known that the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ is a Fréchet space and also that the space of test functions $\mathcal{D}(\mathbb{R}^n)$ is dense in $\mathcal{S}(\mathbb{R}^n)$. Let ...
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0answers
21 views

Distribution and positive measure

Let $f$ be a real fuction, non negative, non decreasing and satisfying $\displaystyle \lim_{x \to - \infty} f(x) = 0 $ and $\displaystyle \lim_{x \to + \infty} f(x) = 1 $. How can I show that the ...
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0answers
9 views

A question of hypoellipticity.

Suppose we have a poly ($Nth$ order) $P$ and that there exists $C$ and $\delta>0$ such that the following inequality holds for all suitable indices $\alpha$: ...
0
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1answer
18 views

Show that $f$ not define tempered distribution

Let $f(x)=\exp(|x|²)$. How can show that $f\in \mathcal{D}'(\mathbb{R}^n)$ but $f\notin \mathcal{S}'(\mathbb{R}^n)$
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1answer
49 views

Distribution with compact support

In my book I have seen that if I have a disitribution $T$ in $D'$ (continuous linear functional on the space $D$) which can be defined for every $C^{\infty}$ function (no necessarily compact supported ...
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2answers
51 views

$\langle u,\phi\rangle=0$ for all $\phi$ implies $u=0$

Proving the injectivity of a function, I arrived to $\forall \phi\in H^s \int_{\mathbb{R}^d} u(x)\phi(x)dx=0$, where $u\in H^{-s}$. I know that if $\phi\in S$ and $u\in S'$, by the definition of the ...
3
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1answer
21 views

Sketch of proof: D (Cinf compact supp) dense in H^s

Let $s$ be any real number. $D$ is dense in $H^s$. $D$ is the space of $C^\infty$ functions with compact support. $u\in H^s$ means, by definition of this space of functions, $\hat{u}\in ...
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0answers
163 views

Inclusions continuous space of test functions

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C ...
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1answer
21 views

Am I correct in interpreting the integral of this delta-function graph?

I have $f(x)$ defined as $\delta(x)-\delta(x-\frac{1}{2})$ $\forall x\in(-\frac{1}{6},\frac{3}{4})$. Outside this boundary it is periodically repeating. From the definition of the $\delta$-function, ...
2
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1answer
40 views

Equivalence between Dirac delta of a function to a usual Dirac delta [duplicate]

Let $f_1(s,\tau)=\delta(e^{(\tau+s)} \sinh \tau )$. This should be equal to $0$ everywhere unless $\tau=0$, but I think there should be some constant multiplying the delta, i.e. it should be ...
0
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1answer
35 views

Is this enough to show this is the delta distribution? [closed]

Let $f$ be a probability distribution with $$\int xf(x) = 0, \quad \int x^2f(x) = 0$$ Clearly, Dirac's delta distribution is a candidate for $f(x)$. Is the previous constrain enough to prove that it ...
0
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1answer
56 views

How to prove that a Schwartz function belongs to $L^p$?

If I have a function $f$ belongs to the Schwartz space, i.e. $f\in \mathcal{S}$, how can I prove $f\in L^p$ ? I know that $\mathcal{S}\subset L^p$ hence the above should make sense. But I need a ...
0
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1answer
24 views

The definition of reflection $\phi(-x)$ for distributions

Define $\tilde{\phi}=\phi(-x)$ and $\langle\tilde{f},\phi\rangle=\langle f, \tilde{\phi}\rangle$. Show this definition is consistent for distributions defined by functions. This is a question ...
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1answer
16 views

The (easy to check?) continuity of an inclusion between Lebesgue-Bochner spaces.

Let $A$, $B$ and $X$ be Banach spaces such that $A$ and $B$ are reflexive; $A\subset X$ with a compact injection; $X\subset B$ with a continuous injection. For $1<a,b<\infty$, set ...
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0answers
11 views

What is the definition for a multivariate function to be well behaved at infinity?

We say a univariate generalized function is well behaved at infinity if it could be approximated by a finite linear combination of finite product of well behaved base functions, like power, sin, cos, ...
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4answers
64 views

Dirac delta property: $f(x)\delta(x-x_0) = f(x_0)\delta(x-x_0)$

Suppose you want to prove that $$ f(x)\delta(x-x_0) = f(x_0)\delta(x-x_0) $$ In my homework, I was instructed to show that the integral of both sides of the equations will lead to the fact that the ...
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2answers
82 views

Dirac distribution verifies $x\delta(x)=0$

Let $\phi:\mathbb R \to \mathbb R$ be a test function. We denote $D(\mathbb R)$ the set of test functions. The dirac distribution $$\delta :D(\mathbb R)\to \mathbb R$$ is defined by: $$<\delta , ...
0
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1answer
50 views

Uniform convergence in the proof of properties of mollifier (Evan's approach)

I am still trying to understand Evans' proof on the properties of mollifier. In the proof of (iii), I understand that the crux of the proof is that uniform continuous function on compact set is ...
0
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1answer
42 views

Schwartz functions have finite $L^p$ norm

It is known that the Schwartz space is dense in $L^p$. And I was told that Schwartz functions are bounded in $L^p$. Could anyone show me "Every Schwartz function is bounded in $L^p$" by explicitly ...
1
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0answers
17 views

Sum of normed function spaces equipped with two-parameter family of norms, Dual inequality

Let $(\cdot,\cdot)$ be the $L^2(\Omega)$ scalar product, and let $V=L^1(\Omega)$, $W=H^{-1}(\Omega)$ (the dual space of $H_0^1(\Omega)$). My question is if there exists a constant $C$, such that for ...