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).
3
votes
1answer
32 views
Definition of convergence in $C^\infty(\Omega)$
Apparently, some of you will count it a silly question; but I am really not convinced or understand, the way they define convergence and then topology as a consequence of convergence.
$\Omega$ is ...
0
votes
1answer
39 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 $n$ ...
0
votes
1answer
62 views
Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$
Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
1
vote
0answers
22 views
Gauss–Ostrogradsky formula for Distributions
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
9
votes
3answers
154 views
Dirac Delta or Dirac delta function?
Is Dirac delta a function? What is its contribution to analysis?
What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
0
votes
1answer
81 views
+50
What will be the support of the convolution of two test functions.
If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$.
Regularization of $g$ is ...
2
votes
1answer
42 views
Regularizing a solenoidal vector field $u\in L^p(\Omega)^N$.
Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and suppose that $u\in L^p(\Omega)^N$, $p\in (1,\infty)$. Assume that in the sense of distributions, $\operatorname{div}u=0$ where ...
4
votes
4answers
100 views
what is the relation of smooth compact supported funtions and real analytic function?
What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
1
vote
3answers
59 views
How can you show that $\delta′=f(0)\delta′−f′(0)\delta$ for a function f that is infinitely differentiable?
Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.
I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation ...
1
vote
2answers
28 views
Find triple functions $ (g_0,g_1,g_2)$ such that $g_0+g_1'+g_2'' = \delta_0-\delta_1$
I want to find a triple of compactly supported continuous functions $ (g_0,g_1,g_2)$ on $\mathbb{R}$ such that $$g_0+g_1'+g_2'' = \delta_0-\delta_1$$
This is seemingly not so hard but ive broken my ...
0
votes
0answers
22 views
Show that$ a$ is a differential of order $m$.
Lat $a = a(x,\zeta) \in S_{1,0}^m(\mathbb{R}^n,\mathbb{R}^n)$. Write $n=n_1+n_2$ with $n_2\geq 1$ and $\zeta = (\zeta_1,\zeta_2)$ with $\zeta_i\in \mathbb{R}^{n_i}$. Suppose that $a$ does not depend ...
1
vote
1answer
43 views
Distribution $P_{\frac{1}{x^3}}$
How to show that for $\phi\in D(R)$, $<P_{\frac{1}{x^3}},\phi>=v.p.\int_{-\infty}^{\infty}\frac{\phi(x)-x\phi'(0)}{x^3}$ defines a distribution?
It is easy to check that $P_{\frac{1}{x^3}}$ ...
5
votes
2answers
44 views
How do we define the $L^p$ norm of a tempered distribution?
I am finishing up a class on function theory and I am trying to reconcile
a few statements that I have seen.
Let us define $L^p(\mathbb R^n)$ to be the set of measurable functions $f$ so that
...
0
votes
1answer
50 views
A basic question about $\operatorname{supp}f$ (support of f).
Is it true that $\operatorname{supp}f$ is the complement of the biggest open set where $f=0
$?
Here $\operatorname{supp}f=$ {$x\in \Bbb R^n ; f(x)\not=0$} and $f\in C$ (collection of continuous maps ...
2
votes
1answer
30 views
Question about distributions
How to show that function $\phi\in D'(R)$ can be represented as derivative of function $\phi_1\in D(R)$ if and only if: $\int_{-\infty}^{\infty}\phi(x)dx=0$.
One direction is easy: if $\phi=\phi'_1$, ...
2
votes
1answer
29 views
Finding distributional limit
How to find $\lim_{\varepsilon\rightarrow 0+}f_{\varepsilon}$ in $D'(R)$, if $f_\varepsilon$ is defined as:
$f_\varepsilon(x)=\frac{1}{\varepsilon^3}$ for ...
4
votes
1answer
103 views
When can you integrate a derivative?
Let us say I have an expression
$$\frac{d}{dt}f = g$$
where the derivative is taken in a weak sense (of distributions). Can I integrate this from $0$ to $t_0$ and get
$$f(t_0) - f(0) = ...
1
vote
1answer
33 views
Estimate derivatives in terms of derivatives of the Fourier transform.
Let us suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a smooth function. Furthermore, for every $\alpha$ multi-index, there exists $C_\alpha > 0$ such that
$$
|D^\alpha f(\xi)| \leq ...
0
votes
1answer
34 views
A clarification about BV functions.
From definition, a locally integrable $ u \in BV(\Omega) $ if its distribution derivative is given by a signed Radon measure. That is there exists $ \mu $ such that for any $ \phi \in ...
2
votes
2answers
86 views
Show existence of a continuous $k$ on $\mathbb{R}^2$ such that $(u,\phi)= \int_{\mathbb{R}^2}k\phi dx$ for all $\phi$
(b). Let $u$ be a distribution on $\mathbb{R}^2$. Assume there exists a continuous function $h$ on $\mathbb{R}^2$ such that $(u,\Delta \phi) = \int_{\mathbb{R}^2}h\phi dx $ for all $\phi\in ...
2
votes
1answer
52 views
Showing that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$
I want to show that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$
$C^0$ is the space of continuous functions, and $H_{\text{loc}}^2(\mathbb{R}^2)$ the set of distributions $u\in ...
2
votes
1answer
39 views
Asymptotic behaviour of solutions to elliptic PDE
Let $u$ be a solution (in the distributional sense) of
$$
\Delta u = \delta_r
$$
on $\Omega \subset \mathbb{R}^2$ open, $r \in \Omega$.
Let $w$ be a solution of
$$
Aw = \delta_r
$$
where
$A = ...
1
vote
0answers
17 views
Solve $P(\frac{d}{dx})u=f$ when $f$ is a distribution with compact support
I need some help for the following statement
Let $P$ be a polynomial, and $P(\frac{d}{dx})u=f$, where $f$ is a distribution with compact support. Then it has a distributional solution $u$ with ...
5
votes
1answer
62 views
Taylor series and tempered distributions
Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid?
When we interpret ...
1
vote
2answers
27 views
Finding limit in $D'(R)$
Find a limit when $\varepsilon\rightarrow +0$ of the following distribution:
$$f_\epsilon=\frac{\varepsilon x}{(x^2+\varepsilon^2)^2}$$
I tried to solve this by putting $x=\varepsilon t$, but ...
1
vote
0answers
31 views
Can we say approximation by smooth function is an equivalent form of “Weierstrass approximation” theorem in Sobolove?
As I came to know most of properties characterized by approximation by smooth functions in Sobolev space looks equivalent to that of Weierstrass approximation theorem in the space of continuous ...
3
votes
0answers
60 views
Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples
Let $f\in L_{loc}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity:
$\hat{f}=\Sigma_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$
With some ...
1
vote
2answers
61 views
Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$
Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$
(a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
2
votes
1answer
56 views
$\phi$ has compact support in $\mathbb{R}^n$ does not imply $\phi (\xi + \eta)$ has compact support in $\mathbb{R}^n\times\mathbb{R}^n$
Let $\phi$ be a $C^\infty$ function with compact support in $\mathbb{R}^n$. Some introductory books on distribution theory I'm reading say that the function $(\xi,\eta)\mapsto \phi (\xi + \eta)$ not ...
2
votes
1answer
46 views
convolution-distributions
We denote by $E'(\mathbb{R})$ the set of distribution with compact support , and $\mathcal{D}(\mathbb{R})$ is the set of function $\mathcal{C}^{\infty}$ with a compact support.
1) I want to compute ...
0
votes
1answer
59 views
convolution-distribution
i want to compute the product of convolution $1 * (\delta' * H)$ where $\delta$ is distribution of Dirac and $H$ is function of Heaviside.
first, we compute $\delta' * H.$ We have by definition that ...
1
vote
1answer
53 views
Sum over cosines = dirac delta - how to get the coefficients?
Given this formula:
$$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$
Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$?
I googled and searched all kinds of ...
0
votes
1answer
35 views
Operation on distributions
I'm currently studying a course on Advanced Real Analysis for a master degree, and our professor handed to everyone of us a 40-page book. I'm major in Algebra, so I'm not really comfortable with this ...
1
vote
1answer
38 views
Limit in a sense of distributions
How to find $\lim_{a\rightarrow\infty} f_a$ in $D'(R)$, for $a>0$, where $f_a:R\rightarrow R$ is defined by
$f_a(x)=\begin{cases}\frac{\sin{ax}}{x}&x\neq 0 \\0&x=0\end{cases}$
Thanks in ...
1
vote
1answer
19 views
Question about convergence in $\mathcal D(\Bbb R)$
Let $\phi\in \mathcal D(\Bbb R)$. How to prove or disprove convergence of $\phi_n(x)=\frac{1}{n} \phi(nx)$ in $\mathcal D(\Bbb R)$?
I tried to do this by definition (we have to check two conditions ...
2
votes
0answers
57 views
Inversion formula for Schwartz-space $\mathcal{S}$.
Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and ...
2
votes
0answers
39 views
Distributions - please check my solution
I have to find a derivative in a distributional sense of the following function (known as Cantor's singular function)
$$f(x)=\left\{
\begin{array}{l l l l l}
0, & \quad\text{$ x\leq 0 $}\\
1, ...
0
votes
1answer
61 views
Show that $\delta(\xi-x)=\delta(x-\xi)$
How would you show $\delta(\xi-x)=\delta(x-\xi)$ if you know that
$$\int _{-\infty}^{\infty}\delta(x)h(x)=h(0)$$
Also how would you then show more generally that if $f(\xi)$ is a monotonic ...
1
vote
1answer
40 views
Find $r(x)$ such that $r(x)L$ is self-adjoint
The differential operator
$$L=a(x)\frac{d^2}{dx^2}+b(x)\frac{d}{dx}+c(x)$$
is not self adjoint. How would you find r(x) such that r(x)L is self adjoint.
I know that this is self adjoint when $L=L^*$ ...
3
votes
4answers
85 views
delta functions $e^{x}\delta (x)=\delta (x)$
How would you prove that;
$$e^{x} \delta (x)= \delta (x)$$
Is it anything to do with the following relationship;
$$ \int_{-\infty}^{\infty} g'(x)h(x)\,dx = \int_{-\infty}^{\infty} g(x)h'(x)\,dx.$$
...
0
votes
1answer
112 views
Is it a dirac-delta?
Hoi, consider $\displaystyle u= \frac{1}{|x|}e^{-|x|}$ for $x\in \mathbb{R}^3$, then one can see
that $\Delta u = u$ for $|x|>0$ ( which one can see by transferring $u$ to spherical coordinates).
...
1
vote
1answer
18 views
Even distribution?
What exactly does it mean for a distribution $u\in \mathcal{D}'(\mathbb{R})$ to be even? Does it mean that for even testfunctions $\varphi $ it holds that $\langle u, \varphi |_\mathbb{R_+} \rangle = ...
1
vote
1answer
61 views
Inverse fourier transform 3 dimensions
Hoi, I want to show that for $n=3$ that $$\mathcal{F}^{-1}\left(\frac{1}{1+|s|^2}\right) = \frac{1}{4\pi |x|}e^{-|x|} $$
As a hint I've been given: Its the unique solution to the equation ...
1
vote
2answers
23 views
Show that the order of $\delta'$ is one
I try to show that the order of $\delta'$ is one. Clearly $|(\delta', \phi)| = |\phi'(0)| \leq \sup \phi'(x)$ but if I got the definition of order right I have to show that $|\phi'(0)| \leq \sup ...
1
vote
1answer
33 views
Multiplication of distributions by smooth functions
Let $u\in D'(\mathbb{R})$ and $f\in C^{\infty}$. I'm trying to figure which of the following statements is true:
I. If $f\restriction_{supp(u)}=1$ then $f\cdot u=u$.
II. If ...
8
votes
1answer
71 views
Various kinds of derivatives
Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$.
Classical derivative. The unique function $f'_c$ defined pointwise by ...
1
vote
1answer
60 views
Calculation of the Laplacian of a function in $\mathbb{R}^3$.
I have to calculate the Laplacian distributional sense) of the following function
$$\frac{e^{i\sqrt{\lambda}|x|}-1}{4\pi|x|}$$
with $\lambda>0$, $x\in\mathbb{R}^3$. I've procedded in the following ...
2
votes
1answer
49 views
Extending a distribution continuously to $C_c^N (\Omega)$
Let $\Omega \subseteq \mathbb{R}^n$ be a domain and let $u\in D'(\Omega)$ be a distribution of order $\leq N$.
How can we show that $u$ can be continuously extended to $C_c^N(\Omega)$?
By ...
2
votes
0answers
73 views
Show that the Dirac delta distribution can not be represented by a continuous function
How do I show that the Dirac delta distribution cannot be represented by a continuous function?
My try is to show that there exists no continuous function $f(x)$ such that
$\int f(x) \phi(x) dx = ...
0
votes
0answers
61 views
Generalized Malgrange-Ehrenpreis Theorem
When can one expect a classical solution of a PDE?
How to use the Malgrange-Ehrenpreis-Theorem
Inspired by the posts above.
The theorem says:
we know if operator $P$ is a polynomial with ...
