Tagged Questions
3
votes
1answer
68 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
53 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 ...
2
votes
1answer
58 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
49 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
56 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?
2
votes
1answer
40 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
31 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 $H^s$?
1
vote
2answers
52 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
64 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
2answers
88 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 ...
0
votes
1answer
68 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 ...
9
votes
3answers
182 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.
1
vote
2answers
34 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
23 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 ...
5
votes
2answers
56 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
...
4
votes
1answer
112 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
$$f(t_0) - f(0) = ...
0
votes
1answer
36 views
A clarification about BV functions.
From definition, a locally integrable $ u \in BV(\Omega) $ if its distribution derivative is given by a signed Radon measure. That is there exists $ \mu $ such that for any $ \phi \in ...
2
votes
2answers
89 views
Show existence of a continuous $k$ on $\mathbb{R}^2$ such that $(u,\phi)= \int_{\mathbb{R}^2}k\phi dx$ for all $\phi$
(b). Let $u$ be a distribution on $\mathbb{R}^2$. Assume there exists a continuous function $h$ on $\mathbb{R}^2$ such that $(u,\Delta \phi) = \int_{\mathbb{R}^2}h\phi dx $ for all $\phi\in ...
2
votes
1answer
52 views
Showing that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$
I want to show that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$
$C^0$ is the space of continuous functions, and $H_{\text{loc}}^2(\mathbb{R}^2)$ the set of distributions $u\in ...
2
votes
1answer
40 views
Asymptotic behaviour of solutions to elliptic PDE
Let $u$ be a solution (in the distributional sense) of
$$
\Delta u = \delta_r
$$
on $\Omega \subset \mathbb{R}^2$ open, $r \in \Omega$.
Let $w$ be a solution of
$$
Aw = \delta_r
$$
where
$A = ...
5
votes
1answer
67 views
Taylor series and tempered distributions
Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid?
When we interpret ...
3
votes
0answers
74 views
Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples
Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity:
$$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$
...
1
vote
2answers
64 views
Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$
Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$
(a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
2
votes
1answer
50 views
convolution-distributions
We denote by $E'(\mathbb{R})$ the set of distribution with compact support , and $\mathcal{D}(\mathbb{R})$ is the set of function $\mathcal{C}^{\infty}$ with a compact support.
1) I want to compute ...
0
votes
1answer
61 views
convolution-distribution
i want to compute the product of convolution $1 * (\delta' * H)$ where $\delta$ is distribution of Dirac and $H$ is function of Heaviside.
first, we compute $\delta' * H.$ We have by definition that ...
2
votes
0answers
63 views
Inversion formula for Schwartz-space $\mathcal{S}$.
Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and ...
0
votes
1answer
118 views
Is it a dirac-delta?
Hoi, consider $\displaystyle u= \frac{1}{|x|}e^{-|x|}$ for $x\in \mathbb{R}^3$, then one can see
that $\Delta u = u$ for $|x|>0$ ( which one can see by transferring $u$ to spherical coordinates).
...
1
vote
1answer
67 views
Inverse fourier transform 3 dimensions
Hoi, I want to show that for $n=3$ that $$\mathcal{F}^{-1}\left(\frac{1}{1+|s|^2}\right) = \frac{1}{4\pi |x|}e^{-|x|} $$
As a hint I've been given: Its the unique solution to the equation ...
1
vote
1answer
37 views
Multiplication of distributions by smooth functions
Let $u\in D'(\mathbb{R})$ and $f\in C^{\infty}$. I'm trying to figure which of the following statements is true:
I. If $f\restriction_{supp(u)}=1$ then $f\cdot u=u$.
II. If ...
1
vote
1answer
60 views
Calculation of the Laplacian of a function in $\mathbb{R}^3$.
I have to calculate the Laplacian distributional sense) of the following function
$$\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
53 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 ...
1
vote
0answers
69 views
Dual space of L_1
EDIT - I was wrong, it turned out that all I need was the Reisz Representation theorem.
I am looking for a way to understand how $L^{\infty}$ is a realization of the strong dual of $L^1$, but I am ...
0
votes
1answer
30 views
Showing $H^1(I) \subset C^{1/2}(\overline{I}) $
Hoi, let $H_1(I)$ the sobolev space on the interval $I = (\alpha,\beta)\subset \mathbb{R}$
http://en.wikipedia.org/wiki/Sobolev_space See here for more on sobolev spaces.
$H_m(I)$ contains all ...
0
votes
1answer
31 views
Boundedness of operator
I want to show that the following Linear operator $L$ is bounded and surjective:
$L: H^2(I) \to \mathbb{C}^4$ where $I = [\alpha,\beta]\subset \mathbb{R}$
given by $L(u) = ...
2
votes
1answer
52 views
Restrictions of distributions
Is it possible to have a distribution $u\in \mathcal{D}'(\mathbb{R})$ such that the restriction of $u$ to $(0,\infty)$ is $\frac{1}{x}$ and the restriction of $u$ to $(-\infty,0)$ is $0$?
It seems to ...
0
votes
1answer
86 views
Is this integral 0?
Let $\phi \in C_0^{\infty}(\Omega)$ with $\Omega = (0,1)\times(0,1)$. Let $u\in L_2(\Omega)$ defined by $u(x,y) = 1$ for $x>y, u(x,y) = 0$ for $x\leq y$
Is there a way we can conclude ...
1
vote
1answer
61 views
Show $A$ is self-adjoint and $ f= Au$ in weak sense.
Hoi, consider $L_2(\Omega)$ with $\Omega = (0,1)\times (0,1)$ and let $u\in L_2(\Omega)$ be defined as
$u(x,y) = 1$ for $x>y$ and $u(x,y) = 0$ for $x\leq y$. Let $A = \partial_x^2 - \partial^2_y$
...
0
votes
1answer
33 views
Identifying 2 spaces of distributions
Hoi,
I want to show that the space $C^{\infty}(\Omega)'$ of continuous linear functionals on $C^{\infty}(\Omega)$ can be identified with the subspace $\mathcal{E}'(\Omega)$ (distributions with ...
1
vote
1answer
40 views
Inequalities for point distribution
Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with
...
1
vote
1answer
50 views
A question on locally integrable function
Let $f$ be a locally integrable function, $f\in L_{\operatorname{loc}}^{1}(\mathbb{R}^n)$. Prove that the operator $$T_f:\phi\to\int_{\mathbb{R}^n}f(x)\phi(x)dx$$
is a distribution.
(See ...
2
votes
2answers
65 views
distribution with point support
Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with
...
1
vote
1answer
108 views
Derivative in the sense of distributions
I have a question regarding calculating the derivative in the distribution sense of the following function:
$$
f(x) = \frac{d^2 }{d x^2}|\cos|x||
$$
Maybe someone can point me in the right ...
4
votes
1answer
137 views
what do test function mean?
I am trying to learn weak derivatives. In that we call $\mathbb{C}^{\infty}_{c}$ function as test function and we use this function in weak derivatives. I want to understand why these are called test ...
1
vote
1answer
55 views
differential equation with distributions
I'm stuggeling with this differential equation:
$T'+T=0$
Where $T$ is distribution.
I found solutions in form:
$\sum_{n\in A} \frac{d^n}{dx^n}\Lambda_{c_n e^{-x}}$. This can be simplified to ...
3
votes
0answers
74 views
Distributional differential equation, somehow related to compact support distributions
I've been mulling over a problem from Friedlander's Introduction to Distribution Theory for a few days now: in Chapter 3 (on distributions with compact support), it asks to solve the differential ...
3
votes
2answers
112 views
approximating Dirac delta with bounded derivatives
Consider the Dirac delta distribution $\delta$ in $\mathbf{R}^d$. It is quite standard to approximate it by functions $g_n$ with $\|g_n\|_{L^1} = 1$.
Is it possible to choose a sequence of test ...
4
votes
1answer
112 views
Generalizing the weak derivative
I am wondering about the weak derivative in time. We say f has a weak derivative f' if $$\int_0^T f\phi' = -\int_0^T f'\phi$$
for all $\phi \in C_0^\infty(0,T)$.
This definition uses the $L^2$ inner ...
1
vote
1answer
64 views
How to prove the density result？
How to prove that $C_c^\infty(R^n)$ is dense in $C^0(R^n)$,where the topology of $C^0(R^n)$ defined as follows
$u_k\rightarrow 0$ in $C^0(R^n)$ if and only if $\forall K\subset\subset R^n$,$sup_{x\in ...
3
votes
1answer
66 views
Equicontinuity and uniform boundedness for “distributions”
Exercise. (Rudin, Functional Analysis, chapter 2, pag. 53). Let us consider the space
$$
\mathcal D :=\{f \in C^{\infty}(\mathbb R), \, \text{supp}f\subseteq [-1,1] \}
$$
with the topology induced by ...
1
vote
1answer
97 views
weak derivative and continuous functions (functionals, distributions)
Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (it vanishes at $t=0$ and $t= T$), and $f \in C^1(0,T \times \Omega)$. Let $w \in L^2(0,T;H^1(\Omega)$ with ...
