Use this tag for questions about Schwartz distributions, also known as Generalised 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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7
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2answers
235 views

Distributions on manifolds

Wikipedia entry on distributions contains a seemingly innocent sentence With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any ...
7
votes
1answer
119 views

Fourier transform using principal value

Can anyone help me compute the Fourier transform of $ 1/|x|^{n-\alpha} $ in $\mathbb{R}^n $ where $ 0 < \alpha < n $ ? Somehow it becomes the principal value of $ 1/|x|^\alpha $ which I can't ...
1
vote
1answer
37 views

Small arguments of $f$ vs large arguments of $\widehat{f}$

Say I know the behavior of $f:\mathbb{R} \to \mathbb{R}$ in the vicinity of 0. Are there any results linking that to the behavior of its Fourier transform $\widehat{f}(\xi )$ for large values of $|\xi ...
2
votes
1answer
72 views

$ \lim_{ \varepsilon \rightarrow 0^+ } \int_{|x| \geq \varepsilon} \frac{ \varphi(x) }{x}dx = - \int_{-\infty}^\infty \phi'(x) \ln(|x|) dx$

How do I prove that $$ \lim_{ \varepsilon \rightarrow 0^+ } \int_{|x| \geq \varepsilon} \frac{ \varphi(x) }{x}dx = - \int_{-\infty}^\infty \phi'(x) \ln(|x|)dx $$ for all $ \varphi \in C_0^{\infty} ...
3
votes
1answer
115 views

How to compute this distribution?

My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
1
vote
1answer
56 views

Power law representation of the delta distribution

In a paper I am reading, the authors claime that by invoking the power law representation of the $\delta$ distribution this gives $$ \delta(k) = (1/4\pi)\lim_{\tau\rightarrow 0} \tau k^{-3 + \tau} $$ ...
1
vote
2answers
252 views

Find the second distributional derivative of $|\sin(x)|$

I am trying to find the second distributional derivative of $f(x)=|\sin(x)|$, where $x$ defined in $\mathbb R$. I have not got far but I started like this: Let $\varphi$ be defined in $S(\mathbb ...
2
votes
1answer
120 views

Distributions as derivatives

Please help me to understand Step (2),(3),(4) and (6) of this theorem. I know mean Value theorem of calculus and here in first step they use it. This is something that mvt implies and I don"t know how ...
2
votes
1answer
72 views

Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?

I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in ...
1
vote
0answers
26 views

Multiplication $ H\times P(1/x) $ in sense of distributions

If $ P(1/x) $ means the principal value of $1/x$ and $ H(x) $ is the Heaviside step function is this then correct (regularization) ...
1
vote
0answers
24 views

Shothostsky's formula regularization

let be the integra $$ I(a)=\int_{0}^{b} dx \frac{f(x)}{(x+ie-a)} $$ with 'e' an small quantity going to 0 from Shothotsky's formula $$ (x+ie-a)^{-1}= P(1/(x-a))-i\pi \delta (x-a) $$ so $$ ...
8
votes
2answers
165 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
2
votes
1answer
144 views

Distribution with finite support.

If $f \in\mathcal D′(\Bbb R^n)$, is a distribution and support of this distribution is a set of finite points in $\Bbb R^n$. Can anyone tell me that what will be the general form of this ...
3
votes
1answer
79 views

Differential equation on $\Bbb R$

We have a differential equation on $\Bbb R$ of the form $$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$ where $\chi_{[0,1]}$ is the characteristic function of the interval $[0, 1] ⊂ \Bbb R$. I want to find a ...
1
vote
0answers
32 views

convergent sequence in quadratic mean and distrubutional sense?

"Say if the following sequence of functions in R: f_n(x) = { 0, if |x| < n; exp(−|x|/n), if |x| > n. converges (1) in quadratic mean, (2) in the sense of distributions." My own calculation of (1) ...
2
votes
1answer
63 views

Find the distribution $T$ defined in $\mathcal S'(\mathbb R)$

Let $\mathcal S'$be the space of all tempered distributions. Find a distribution $T \in\mathcal S'(\mathbb R)$ such that $xT = \sin^2(x) $, and moreover $T(\phi) = \pi$ for $\phi(x) = \exp(−x^2)$.
2
votes
1answer
273 views

function a.e. differentiable and it's weak derivative

Note - I am just starting to learn about theory of distributions, so this may be a trivial question, if so I'd be grateful for a reference, nevertheless the question is the following: suppose I have a ...
0
votes
1answer
65 views

Question about distribution $F_h=\frac{f(x+h)-f(x)}{h}$

How to solve the following: Let $f:R\rightarrow R$ be a continuous function, $h\in R-\{0\}$ and $F_h:R\rightarrow R$ defined with $F_h=\frac{f(x+h)-f(x)}{h}$. Prove that a)$F_h\in D'(R)$, ...
6
votes
0answers
229 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
3
votes
2answers
182 views

Fourier transform

I am new to the distribution theory and have some difficulties to calculate curtain fourier transforms. Can you help me with $$\frac{e^{-xb}}{x+i0}$$ I got to the point $$\lim_{\epsilon \to ...
2
votes
0answers
89 views

Fourier transform of a tempered distribution

anybody knows how to calculate the fourier transform of $e^{-ax}, a>0$ in the sense of tempered function. I manage to find out it is $ \delta $ (y+ia) but it does not seem right as the 'argument' ...
4
votes
1answer
172 views

Solution of a differential equation in space of distributions

I cant figure out how to find general solution to equation $(1+x)^2 u''=0$ in the space of distributions. Any ideas?
1
vote
1answer
96 views

Solving the equation $\nabla u=f$.

Let $\Omega\subset\mathbb{R}^N$ be a open set, $\mathcal{D}(\Omega)=C_0^\infty(\Omega)$ and $D'(\Omega)$ the set of distributions. Suppose that $f_i\in \mathcal{D}'(\Omega)$ for $i=1,\ldots,N$. Define ...
2
votes
1answer
53 views

Finding limits in $D'(R)$

How to find the following limits in $D'(R)$: $\lim\limits_{k\rightarrow \infty} \cos{kx} P_{\frac{1}{x}}$ and $\lim\limits_{a\rightarrow 0+} e^{ax} P_{\frac{1}{x}}$ ($k\in N$, $a\in R$). I tried ...
3
votes
0answers
44 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha ...
5
votes
0answers
82 views

Show that integrals are equal

Let $f \colon [0,+\infty) \to \mathbb R$ be a convex function. Then $f''(x)$ is a nonnegative distribution on $(0,+\infty)$ and hence it can be continued to a nonnegative mesure $\mu$ on ...
4
votes
1answer
120 views

Compatibility of pointwise and distributional convergence

This has probably been asked before but I couldn't find it. Let $\Omega$ be an open subset of $\mathbb{R}^n$ and let $u_k,\, u \in L^1_{\mathrm{loc}}(\Omega)$ and $v\in \mathscr{D}'(\Omega)$ (the ...
1
vote
1answer
255 views

Convolution of distributions.

We are given with distributions $f,g \in D'(\Bbb R)$. If $suppf\subset (-\infty,a)$ and $supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution. where $a$ and $b$ are real ...
1
vote
0answers
32 views

Convergence of extensions

Let $v_n\in L^2(\mathbb{R}^{n}_+)$ and suppose it converges to $v\in L^2(\mathbb{R}^{n}_+)$. Furthermore let $v_n$ be such that $(I-\Delta) v_n=0$. Here we interprete ...
1
vote
1answer
47 views

Finding limit in D'(R)

How to find $\lim_{\varepsilon\rightarrow 0+} f_\varepsilon$ in $D'(R)$, if $f_\varepsilon(x)=\frac{\sin{\varepsilon x}}{x}$? Thanks in advance.
2
votes
1answer
114 views

The differentiability of convolutions

Yes, again, this type of question. Similar ones this and this. I come with another variant. Let $f\in\mathcal{S}$, i.e. Schwartz function, and $g\in L^{p}(\mathbb{R}^d),p\in[1,\infty]$. The following ...
5
votes
1answer
89 views

Distribution with singularities.

I need some help to prove that $f$ defined by $\langle f,\psi\rangle:= \sum_{n=0} ^\infty \psi^{(n)}(n)$ is a distribution which has singularities of infinite order. Here $\psi$ is a test function ...
3
votes
2answers
94 views

Problem of convolution.

If we are given with a polynomial $\mathcal P$ and a compactly supported distribution $g$. Can we prove that their convolution will be a polynomial again?
5
votes
1answer
93 views

Uniqueness of Ordinary Differential Equations in $D^{'}$, the space of Schwartz distribuitions

Let $m \in \mathbb{N}$. For $k=1,...,m$ let $a_k : \mathbb{R} \rightarrow \mathbb{C}$ be a $C^{\infty}$ function. And suppose that: $a_m(x) \neq 0 \; \forall x \in [x_0, \infty[$ And let P be the ...
4
votes
1answer
164 views

Local integrability of the convolution of a function with a distribuition

Let $G_n$ be the following distribuitions for $n\geq3$ (for $n=2$ it is just a function) in $\mathbb{R^n}$ (the fundamental solutions of the Laplace equation in $\mathbb{R^n}$ ): ...
4
votes
1answer
244 views

Solving distributional differential equation

How to solve differential equation in $\mathcal D'(R)$: $$u''+u=\delta'(x),$$ where $\delta$ is Dirac Delta function? Solution of homogeneous problem is $C_1\cos{x}+C_2\sin{x}$, so using the ...
1
vote
1answer
108 views

Differential equation - distributions

How to find solution to the following problem (in $D'(R)$): $$u''+3 u=1+\delta (x)\text{ ?}$$ Thanks in advance.
3
votes
1answer
278 views

Second derivative Dirac delta distribution times $(x-a)^2$, intepretation

I'm not sure if this calculation is correct and also if interpret it correctly (from old exam), Show that $ (x-a)^2 \delta ''_a = 2 \delta _a $. We have for distributions $f$ and test functions ...
2
votes
1answer
54 views

Support of a generalized function

Let $F \in D'(\mathbb{R})$. If there exists $f \in D(\mathbb{R})$ which vanishes on $supp(F)$, but $F(f)\neq0$, then how can one prove that $supp(F)$ contains an isolated point? For example, if ...
1
vote
1answer
35 views

Continuity of the extension of a distribution to $H^s$

Let $u\in D'(\mathbb{R}^n)$ be a distribution and suppose that $u$ can be extended to linear functional on $H^s$. Does it follow that $u$ can be extended to a continuous linear functional on $H^s$?
0
votes
2answers
96 views

Dirac delta from polar coordinates to cartesian coordinates

I have: $$k_x = k \cos\theta\\k_y=k\sin\theta$$ I would like to rewrite in terms of $k_x$ and $k_y$: $$\exp(in\theta)\,\frac{\delta(k-\alpha)}{k}$$ I start from: ...
1
vote
2answers
105 views

Relationship of the support of a test function with the support of distribution.

Let $\phi\in D(\Omega)$ and $f\in D'(\Omega)$. If $\phi$ is $0$ in a neighbourhood of $\operatorname{supp}(f)$, then how will we prove that $\langle f, \phi \rangle$ is also $0$? Will it be ...
4
votes
1answer
102 views

A problem from distribution theory.

Let $f$, $g\in C(\Omega)$, and suppose that $f \neq g$ in $C(\Omega)$. How can we prove that $f \neq g$ as distributions? Here's the idea of my proof. $f$ and $g$ are continuous functions, so they ...
3
votes
1answer
87 views

prove $ \mathcal F(f) = c_1\delta + c_2 \delta' + c_3\delta'' + T_g $ with $f(x)=|x^2 -1|$

let $f(x)=|x^2 -1|$ be a tempered distribution (i proved it) , and calculated its 3rd derivation (as a distribution) and then this stopped me : prove that we have : $$ \mathcal F(f) = ...
2
votes
1answer
68 views

prove the existence of $c\in \mathbb C$ such as $\mathcal F(g)=c\delta + {1\over2i\pi}V_p({\mathcal F(f)\over x}) $

prove that it exist $c\in \mathbb C$ such as $\mathcal F(g)=c\delta + {1\over2i\pi}V_p({\mathcal F(f)\over x}) $ with $\mathcal F$ fourier transform , $f\in \mathbb D(\mathbb R)$ , $\delta$ ...
1
vote
1answer
130 views

distribution - exercises

Let $f \in \mathcal{C}(\mathbb{R}) \cap L^1(\mathbb{R}).$ We suppose that $f$ is bounded and even $(f(-x)=f(x))$. let $\lambda \in \mathbb{R}.$ We assume that: $\int_{-1}^1 \dfrac{1 - f(\lambda ...
1
vote
1answer
143 views

Integral with delta Dirac power

Is it possible to calculate the integral: $$J=\int_{-\infty}^{+\infty}f(x)\delta(x-x_0)^kdx$$ wih $k\in\mathbb{R}$? I know that in the Colombeau algebra the distribution $\delta(x)^2$ is defined. What ...
4
votes
2answers
194 views

Definition of convergence in $C^\infty(\Omega)$

I am not convinced or may be don't understand, the way they define convergence and then topology as a consequence of convergence. $\Omega$ is open subset of $\Bbb R^n.$Define standard topology on ...
3
votes
2answers
1k views

Dirac delta in polar coordinates

Given $$x=r\,\cos\theta\\y=r\,\sin\theta$$ and $$x'=r'\,\cos\theta'\\y'=r'\,\sin\theta'$$ how can I express $$\delta(x'-x)\delta(y'-y)$$ in terms of the polar coordinates? And the more general ...
2
votes
1answer
80 views

Multiplication in $\mathcal D'(R)$.

I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...