1
vote
0answers
31 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
6
votes
1answer
48 views

Which normed space have Fatou's property?

There is tool in mathematics, more specifically in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property. Let $E$ be a normed ...
1
vote
1answer
23 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
3
votes
1answer
46 views

The Dirac Delta Distribution $\delta_0 : D \to \mathbb R$ is not regular

I have a question on the following proof, that $\delta_0$ is not a regular distribution. We define $\delta_0$ as the linear function on test function with $$ \delta_0(\varphi) = \varphi(0) $$ for ...
2
votes
1answer
37 views

Distributions with support of the form $\left\lbrace x \right\rbrace$

Doing some calculations with Distributions I came up with the following theorem: THEOREM: Let $O \subseteq \mathbb{R^d}$ be an open subset and $x \in O$. Suppose $T \in \mathcal{D}'(O)$ with ...
3
votes
2answers
72 views

Topology of test functions $\mathcal{D}(\mathbb R)$

(My motivation for the following question is to understand the distribution theory) The space of test functions: $\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb ...
2
votes
1answer
31 views

Duals of embeddings in the space of distributions

If $ \Lambda \colon X \hookrightarrow \mathcal{D}'$ is a continuous embedding of a normed vector space $X$ into the space of distributions (for example $X=L^p$), is it true that the dual of ...
2
votes
0answers
104 views

Derivative of Dirac delta behavior at 0

I calculated the curl of the vector field $\vec j = \delta(z) f(\rho) \vec e_\phi$ in cylindrical coordinates (ρ,φ,z). For the $\vec e_ρ$ unti vector I got: $$-\delta'(z)f(\rho)$$ The main question ...
0
votes
1answer
33 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
2
votes
1answer
51 views

Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to ...
3
votes
1answer
164 views

Topologies of test functions and distributions

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my ...
3
votes
1answer
67 views

Applications of the theory of distributions outside of PDEs?

Are there any interesting, important or powerful mathematical applications to the Theory of Distributions besides those dealing with partial differential equations?
2
votes
1answer
51 views

Weighted Dirac comb as a tempered distribution?

I'm trying to determine when a "weighted" Dirac comb is a tempered distribution. More precisely, trying to prove: $$u=\sum_{k=1}^{\infty}c_k \delta_k\in\mathcal{S}'(\mathbb{R})\iff\exists ...
0
votes
2answers
52 views

A short question concerning the distributional solution of $xf=0$

I was reading my notes on the following result: All the $\mathcal{D}'(\mathbb{R})$ solutions to $xf =0$ are of the form $c\delta $ where $c$ is constant and $\delta$ is the dirac delta distribution ...
2
votes
2answers
59 views

$xT' = 1$ in $D'(\mathbb{R})$

I need help solving the following problem: I want to show that all solutions of $$xT' = 1\ , T \in D'(\mathbb{R})$$ take the following form: $c_{1} + c_{2}1_{[0, \infty)} + ln|.|$ What I tried so far ...
2
votes
1answer
39 views

Translation is continuous

Let $\mathcal D$ be the space of 'test-functions'. Those are infinitely differentiable functions with compact support. Define the following convergence on $\mathcal D$. $(\phi_j) \to \phi$ in ...
2
votes
1answer
376 views

Fourier Transform of Dirac Comb on $\mathbb{Z}$ and $\mathbb{Z}^{d}$.

Let $f(x)=\sum_{n\in\mathbb{Z}}\delta(x-n).$ (a) Show $f$ is a tempered distribution. (b) Compute $\hat{f}$ using the convention $\int_{\mathbb{R}}f(x)e^{-ix\xi}\;dx$ convention for $\mathcal{F}$. ...
3
votes
2answers
81 views

Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
0
votes
2answers
81 views

What is the closure of $ C^\infty_c(\mathbb{R}^n\setminus\{0\})$ in Sobolev $ W^{1,p} $ norm?

For $1 \leq p < \infty, n\geq 1 $ my guess of the answer was $ W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $ \overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq ...
0
votes
0answers
30 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
2
votes
1answer
129 views

Convergence of a integral - heat Kernel and dirac delta function

Consider $\varphi \in S(R^n)$ (space of rapidly decreasing functions). Consider the heat kernel $$ K_t(x) = \displaystyle\frac{1}{{(4\pi t)}^{n/2}} \displaystyle e^{- \displaystyle\frac{|x|^2}{4t}}, ...
0
votes
1answer
25 views

For every $A\in \mathcal{L}(C^\infty(\mathbb T^n))$ exists $k_A\in C^\infty(\mathbb T^n\times \mathbb T^n)$ such that..

does anyone know whether it is true that for every $A\in\mathcal{L}(C^\infty(\mathbb T^n))$ there exists $k_A\in C^\infty(\mathbb T^n\times \mathbb T^n)$ such that, $$ Af(x)=\int_{\mathbb T^n} k_A(x, ...
2
votes
1answer
132 views

$ \frac{\partial^2 T}{\partial x\partial y} = 0 $, then $ T = ? $

Can we characterize all distributions $T \in \mathcal{D}'(\mathbb{R}^2) $ with the following property of distribution derivatives ? $$ \frac{\partial^2 T}{\partial x\partial y} = 0 $$ For functions it ...
0
votes
0answers
39 views

Convolution Kernel..

does anyone what is the definition of the convolution kernel of an operator, $$A: C^\infty(\mathbb T^n)\rightarrow \mathcal{D}^{'}(\mathbb T^n),$$ where $\mathcal{D}^{'}(\mathbb T^n)$ is the space ...
3
votes
1answer
59 views

Schwartz kernel theorem in the case the distributions are induced by smooth functions..

how can I show that if $A:C^\infty(\mathbb T^n)\rightarrow C^\infty(\mathbb T^n)$ is a continuous linear operators then there is a unique linear and continuous operator $K_A: C^\infty(\mathbb ...
0
votes
1answer
22 views

Approximation of Delatadistribution

I'm trying to understand a computation in my physics script. To describe the Deltadistribution $\delta(x) $ correctly we would need the formalism of distributions, but one can also much less ...
1
vote
0answers
81 views

Uniformly convergence of delta sequences

Let $(f_n) $ be a sequence of continuous functions $f_n: \mathbb R \rightarrow \mathbb R$ such that $$ \lim_{n\rightarrow \infty}\int_{\mathbb R} f_n(x) \phi(x) dx=\phi(0) $$ for each continuous ...
0
votes
1answer
105 views

Convergence to $\delta$ distribution

Show that $$v_{t}(x) = (4 \pi kt)^{- \frac{1}{2}} \exp \left( -\frac{a x ^2}{4kt} \right)$$ converges to $\delta_{0}$ in $D'(\mathbb{R})$ when $t \to 0^{+}$. Asumming that: $$\int_{\mathbb{R}} ...
0
votes
1answer
64 views

If $u\in L^p$, is $u\in L^q$ for some $q>p$?

(Motivation is below) Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$. Let $p\in [1,\infty)$ and $u\in L^p(\Omega)$. Is there any $q>p$ such that $u\in L^q(\Omega)$? I already know that ...
4
votes
4answers
6k views

Derivative of a Delta function

I know questions similar to this one have been asked, but there is a particular aspect that I'm confused about that wasn't addressed in the answers to the other ones. I'm dealing with an expression ...
1
vote
0answers
29 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) ...
2
votes
1answer
67 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 $\phi(x) = \exp(−x^2)$.
6
votes
1answer
99 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
172 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 $\mathbb{R^n}$ ): ...
3
votes
1answer
349 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 ...
1
vote
1answer
36 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 $H^s$?
5
votes
1answer
159 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 ...
2
votes
0answers
86 views

Can we say approximation of Sobolev function by a smooth function is inspired from “Weierstrass approximation”? [closed]

As we see that the idea of approximation of a function in Sobolev space by a smooth function looks like an abstraction of Weierstrass approximation theorem..
1
vote
1answer
328 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
78 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
107 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
62 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 ...
0
votes
1answer
98 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 ...
0
votes
2answers
582 views

example of a function with compact support

Can you give an example of a function which is $C^\infty (\mathbb{R})$ having support on (-1,1) such that $ \int_{-1}^1 f(x)\,dx$=1 and $ \int_{-1}^1 xf(x)\,dx$=0. Thank you.
1
vote
0answers
31 views

Function of the incremental ratio tends weakly to a distribution

Let $g:\mathbb{R}^3\to\mathbb{R^2}$ be a continuous function. Suppose that there exists $\Omega$ a neighborhood of $0$ where $Xg, Yg \in L^\infty(\Omega)$, with ...
2
votes
1answer
56 views

A particular product of distributions

Suppose you have two continuous functions $f,g: \mathbb{R}\to\mathbb{R}$; is the product $f'g$ as a distribution, at least locally? I am interested in a local result, actually, so you can as well ...
1
vote
1answer
112 views

The Dirac impulse and Fourier transform

Here wikipedia it is said that the Dirac delta could be thought of as $$ \delta(x) = \left\{ \begin{array}{ll} \infty &, x = 0 \\ 0 &, x \ne 0 \end{array}\right. $$ and here that the ...
2
votes
1answer
205 views

Proof that limit goes to zero without Riemann-Lebesgue lemma

Let $\varphi$ be a test function ($\varphi$ is smooth and has compact support - is zero outside some bounded interval). I know that the following $$ \lim_{\epsilon \to 0_+} ...
1
vote
0answers
61 views

Lebesgue-Stieltjes integral as a generalized function

Given some convex function $f(x)$, $x >0$ we can define a distribution $F \in \mathcal{D}'(0,\infty)$ using Lebesgue-Stieltjes integral $$ \langle F, \varphi \rangle ...
3
votes
1answer
190 views

Second derivative of convex function

Let $f(x)$, $x>0$ be a convex function. Then it's distributional second derivative is defined by the rule $$ \langle f''(x),\varphi(x)\rangle = \langle f(x), \varphi''(x)\rangle $$ for any ...