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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2
votes
1answer
79 views

$\langle u,\phi\rangle=0$ when ${\rm supp}(u)\cap{\rm supp}(\phi)^\circ=\emptyset$?

I know that $\langle u,\phi\rangle=0$ if ${\rm supp}(u)\cap{\rm supp}(\phi)=\emptyset$. For some while I wondered whether it's enough that $\phi$ vanishes on ${\rm supp}(u)$ but that's not true, as ...
1
vote
1answer
72 views

Dirac's delta, infinite series and integral

Why $\int_{-\infty}^x\sum_{i=1}^{+\infty} p^{i-1}\delta(\alpha-i)d\alpha = \sum_{i=1}^{+\infty}\int_{-\infty}^x p^{i-1}\delta(\alpha-i)d\alpha$ where $\delta$ is the Dirac's delta and $p \in ]0;1[$ ...
7
votes
1answer
98 views

Proving that a family of functions limits to the Dirac delta.

For each $\epsilon > 0$, define $f_\epsilon:\mathbb R\to \mathbb R$ as follows: \begin{align} f_\epsilon(k) = \frac{1}{\pi}\frac{\epsilon}{\epsilon^2+k^2}. \end{align} How does one rigorously ...
0
votes
1answer
57 views

two dimensional delta function

Is it correct to write $\delta(x,y)=\delta(x)\delta(y)$ where $\delta(x,y)$ is the delta function in two dimensions? Or are there some cases where the above fails to give the correct results when ...
1
vote
0answers
34 views

Fundamental solutions for wave equations which vanish inside characteristic cones

If the number of dimensions $n$ is odd, the flat-space wave operator $-\partial_t^2 + \nabla^2$ admits the fundamental solution $(-\sigma)^{1-n/2} \Theta (-\sigma)$, where $\sigma = - t^2 + |x|^2$. ...
3
votes
0answers
44 views

Distribution induced by a function

We are given $F(x) = |2x+1|, x \in \mathbb{R}$ How to determine whether $$[F|_{(0, \infty)}] \in \mathcal{D}'((0, \infty))$$ $$[F|_{(- \infty, 0)}] \in \mathcal{D}'((- \infty, 0))$$ are regular ...
0
votes
2answers
28 views

$\text{supp }u =\{0\}$ implies $u=c\delta_0$ in distributional sense?

Given $\text{supp } (u) =\{0\}$ where $u\in D(X)^\prime$ is a distribution and $X\subset \mathbb{R}$. Does this already imply that $u=c \delta_0$? for some constant $c$.
0
votes
0answers
23 views

Any way to rewrite/simplify $\int dp ~f(p)~\delta[c-g(p)-E(p)]$?

Im thinking of using $$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$$ but in my case $$\int dp ~f(p)~\delta[c-g(p)-E(p)]$$ the functions $f$ and $E$ are basically unknown, while $g$ is a ...
3
votes
2answers
58 views

What is the meaning of $1^\lor=\delta$

I tried to look up the definition of a $\lor$ and it does not seem to explain this particular usage $$1^\lor=\delta$$ This is used in a proof that inverse fourier transform of $1$ is $\delta$, but ...
0
votes
1answer
75 views

Inverse of laplacian operator

I recently read a paper, the author treats $$\int_{\mathbb{R}^d}f(y)\cdot \frac{1}{|x-y|^{d-2}}\,dx = (- \Delta)^{-1} f(y)$$ up to a constant in $\mathbb{R}^d$. I am not familiar with unbounded ...
0
votes
1answer
48 views

How do I find the distribution of the laplacian operator acting on Log |f|

Can someone give me some ideas/insight/suggestions on approaching this problem: Calculate the distribution $u(x) = \Delta \log{|\,f\,|}$ where $f$ is a meromorphic function that doesn't vanish ...
0
votes
1answer
65 views

What is ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$?

How exactly can I prove that ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$ equals the linear space $M$ spanned by 'simple fucntions' $\phi\in{\cal D}(\Omega_0\times\Omega_1)$ of the form ...
0
votes
1answer
52 views

Variation of Delta function integrated

We all know: $$\int_0^\infty \delta(y) dy = 1.$$ How about $$\int_{-\infty}^\infty y\delta(y) dy .$$ The solution of this is $0$. I have no idea how to get this. thx,
3
votes
1answer
74 views

Do sequences fully specify the topology of $\mathcal{D}(\Omega)$ and $\mathcal{D}'(\Omega)$?

It is well known that $\mathcal{D}(\Omega)$ and $\mathcal{D}'(\Omega)$ are not metrizable, and that a topological vector space is metrizable if and only if it is first-countable (Rudin, Thm. 1.24). ...
3
votes
1answer
49 views

Convergence of series of integrals

Let $\phi \in C^\infty(\mathbb R)$ be a function such that $\phi(x), \phi'(x) \to 0$ as $x \to \infty$. I want to show that $$\lim_{n \to \infty} \int_\mathbb R \cos(nx) \phi(x) \ dx = 0$$ Doing it ...
1
vote
0answers
21 views

Recognizing regular distributions

By "regular" distributions I understand those Schwartz distributions that arise from locally-integrable functions. Are there ways of telling them apart from the non-regular ones? Does the set of those ...
4
votes
1answer
70 views

Understanding Green's function

I have a problem in understanding the definition of Green's function which occurs when solving a Poisson equation $\Delta u=f$. Here is the definition of our lecture: Let $\Omega\subset\mathbb{R}^n$ ...
1
vote
1answer
27 views

Derivative of a distribution

$\DeclareMathOperator{\vp}{v.p.}$ We define $\vp \frac 1x \in \mathcal D'(\mathbb R)$ (the principal value of $\frac 1x$) as $$\left\langle \vp \frac 1x, \varphi \right \rangle = \lim_{\varepsilon ...
1
vote
1answer
89 views

Intiution behind the derivative of dirac delta function

Let me first begin what I mean by saying the intuition behind the " $\delta'(x)$ ". For example the smooth approximations of the delta function looks like the following: (Left:the smooth ...
1
vote
1answer
40 views

Pre-dual of distributions with support in a closed subset

Usually, in order to define the space of distributions $\mathcal{D}'(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$, one considers the space $\mathcal{D}(\Omega)$ of $C^\infty$-test ...
1
vote
1answer
115 views

How to prove that the Dirac delta is not a function?

I am currently taking a course on test functions and distributions and my task is to prove that the Dirac delta is not a function. Furthermore, I would also like to prove that it is continuous as a ...
3
votes
1answer
43 views

Defining a distribution

We fix the space $\mathcal{D}=\mathcal{C}^\infty_0(\mathbb{R}^n)$ as space of testfunctions. Let $(f_n)$ be a sequence of distributions with $\lim_{n\to\infty} f_n(\varphi)$ existing for all ...
4
votes
1answer
49 views

Differential Equation In Distributions

I need to find all $u \in \mathcal{D}'$ ( space of distributions) such that $ e^x (e^{-x} u ) ' = \delta_0 +1$. For any $\phi \in C_0^\infty(\mathbb{R})$ we have $\langle e^x (e^{-x} u ) ' , ...
-1
votes
1answer
67 views

Fourier Transform of a Temperate Distribution

Let $f$ be a temperate distribution. Suppose that $f$ is a solution to the equation $ f'-f= \delta_0 +1 $. I want to find $ \hat{f}$... Here's what I did: Since $ f'-f= \delta_0 +1 $, then $ ...
3
votes
0answers
43 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
3
votes
0answers
78 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
0
votes
0answers
54 views

Why $\mathcal{D}(\Omega)$ be a topological vector space is important?

Let $\Omega\subset\mathbb{R}^N$ be an open set and $\mathcal{D}(\Omega)$ the set of all infinitely differentiable functions with compact support on $\Omega$. In the study of PDEs, we use the ...
2
votes
1answer
51 views

Show that a distributional solution of $\Delta u = f u $ is smooth for smooth $f$

As in the title - I would like to show that if $f$ is a smooth ($C^{\infty}$) function then for any distribution $u$ satisfying $$ \Delta u = fu$$ in the distributional sense we have, in fact, $u ...
2
votes
1answer
67 views

Delta distribution as a bounded linear functional

Let $\Omega \subset \mathbb{R}^n$ be a domain with $0 \in \Omega$. For which $n \in \mathbb{N}$ is the Dirac-delta distribution a bounded linear functional on $H_{0}^{1}(\Omega)$, that is to say, an ...
1
vote
1answer
65 views

Is the solution of a PDE always the convolution of the Green function?

If we are given Poisson's equation in a space: $$\nabla ^2 u=F$$ The solutions (those who admit Fourier transform) are given by: $$u(x)=\int_\mathbb{R^n} G(x,y)F(y)dy$$ Where $G$ is the Green ...
2
votes
0answers
52 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
0
votes
1answer
20 views

Product of $C^\infty$ and $\mathcal D$

On my book there is the following statement: We can define the product of a distribution $u \in \mathcal D'(\Omega)$ and a function $\psi \in C^\infty(\Omega)$ in the following way: ...
1
vote
1answer
39 views

Distribution, equation

I need to find all solutions in $\mathcal {D'}(\mathbb{R})$( distributions $T$ ) of this equation: $id \cdot T' = 0$ $T'(\varphi ) = - T(\varphi ')$ I've already solved a similar equation $id ...
0
votes
1answer
46 views

Find distributions whose second derivative is Dirac delta

How can I find distributions $T \in \mathcal{D'}(\mathbb{R})$ such that $T'' = \delta _0$ ? I know that $D^2T(\phi) = T(D^2 \phi)$. $\phi \in C^{\infty}(\Omega, \mathbb{R}), supp \ \phi$ is compact ...
0
votes
2answers
40 views

distributional derivative of $x^{-1/2}$.

Consider the distribution defined for $\phi \in C_{c}(\mathbb{R})$ by $$T(\phi) = \int_{-\infty}^{\infty} |x|^{-1/2} \phi(x) dx.$$ Compute its derivative $T^{\prime}(\phi)$. Attempt: I use ...
2
votes
1answer
110 views

For what parameters does a sequence converge in $S$

Let $S$ be space of rapidly decreasing functions $f\in C_0^\infty(\mathbb R^n)$, that for any multi-indices $\alpha$ and $\beta$ there is a constant $M_{\alpha,\beta}$ such that $$|x^\alpha D^\beta ...
3
votes
1answer
94 views

does the function $|\sin(x) |$ define a tempered distribution? if so compute the fourier transform

I need to check if the function $|\sin(x)|$ defines a tempered distribution and find the fourier transform of the distribution. I think it defines it because it is summable on every compact ...
1
vote
1answer
40 views

Distribution as limit of quotient

Let $u \in \mathcal{D}'(\mathbb{R})$. Show that: $$\frac{u-\tau_{x}u}{x} \to \mathcal{D}u$$ in $\mathcal{D}'(\mathbb{R})$ when $x \to 0$. Here, we define $\tau_{s}f(x)=f(x+s)$. It's an exercise ...
0
votes
2answers
41 views

Distributional derivative of $\sin { (\pi|x|/2))} \chi_{\{|x|<2\}}$

I have to find the first and the second distributional derivative of the function : $$f(x) = \begin{cases} \sin { (\pi|x|/2))} \quad & |x| <2 \\ 0 \quad & |x| \geq 2 \end{cases}$$ but ...
0
votes
3answers
83 views

Distributions defined by $C_0^\infty(\mathbb{R})$ enough to distinguish $f_1,f_2\in L^1(\mathbb{R})$?

Let $f_1,f_2$ be Lebesgue-summable functions on the real line. I was wondering whether space $C_0^\infty(\mathbb{R})$ of infinitely differentiable compactly supported functions, intended as ...
0
votes
0answers
42 views

How should I use Sobolev embedding to prove Schwartz representation theorem?

From Richard Melrose, From Microlocal analysis to global analysis, Chapter 1. Show that, for any $p\in \mathbb{R}$ the map $$ R_{p}:S(\mathbb{R}^{n})\ni \phi\rightarrow ...
3
votes
0answers
33 views

What is the topological interpretation of continuity of distributions?

I was given this definition of continuity in the distributional sense. A distribution $T$ over the space of test functions $\mathcal{D}$ is continuous if for every sequence of test functions $\{ ...
4
votes
1answer
113 views

Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Fourier series of a function that belongs to $L^2([0,T])$ can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space $L^2([0,T])$. Fourier transform of a ...
10
votes
2answers
282 views

Is this sequence bounded ? (An open problem between my schoolmates !)

Let $f$ be a smooth function (say $\mathcal{C}^{\infty}$) in its two real variables ($t$ and $T$). I consider the following sequence defined by $$A_n:=\lim_{T \to \infty} \int_{0}^{1} e^{-n t} ...
3
votes
0answers
70 views

What is a good way to prove the weak form of the Sobolev embedding theorem?

From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11. Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
0
votes
1answer
42 views

Proof on delta sequences

In a coursetext of mine there is a theorem that says: when $f(x)$ is locally integrable ($f(x)\in\text{L}^1_{\text{loc}}(\mathbb{R}^d$), $f(x)\geq0 \,\forall x$ and $\int f(x) dx=1$ then it follows ...
0
votes
0answers
30 views

Operator of Taylor series as a distribution?

I would like to prove a statement: $$T:=\sum_{k=0}^\infty a_k\partial_x^k\delta_0\not\in\mathscr{S}'(\mathbb{R}^n)$$ and, in contrast, $$T_n=\sum_{k=0}^n ...
3
votes
2answers
57 views

Correct definition of convolution of distributions?

Wikipedia states, that the definition of convolution of function $f$ with a distribution $T$ is $$\langle T\ast f,\varphi\rangle=\langle T,\tilde{f}\ast\varphi\rangle$$ where $\langle ...
1
vote
0answers
44 views

Represent Dirac distruibution as a combination of derivatives of continuous functions?

It seems from Rudin's functional analysis (P168, Thm6.27), the Dirac distribution $\delta_0$ on $R^1$ can be written as $$ \delta_0=f_0+f'_1+f''_2, $$ for $\{f_i\}_{i=0}^2\in C(R^1)$, there $'$ ...
1
vote
0answers
18 views

exactly represent Dirac distribution as derivatives of continuous function [duplicate]

It seems that in Rudin's functional analysis (P168, Thm6.27) that the Dirac distribution $\delta_0$ on $R^1$ can write as $$ \delta_0=f_0+f'_1+f''_2, $$ for $\{f_i\}_{i=0}^2\in C(R^1)$, there $'$ ...