1
vote
0answers
21 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
2
votes
1answer
36 views

Show that the Fourier transform of a a distribution is $C^{\infty}$

I am trying to understand the solution to the following problem: Let $u \in \mathcal{D}'(\mathbb{R}^{n})$ such that $u(x) = c \log(|x|)$ when $|x|>1$, where $c \in \mathbb{C}$. Show that $u \in ...
1
vote
0answers
45 views

Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a ...
1
vote
1answer
56 views

Find distribution solving a differential equation

I think I have solved the following differential equation, but I am not sure of all steps are justified. Exercise: Find all distributions $u \in \mathcal{D}'(\mathbb{R})$ such that $x(u' -u) = ...
1
vote
0answers
33 views

Finding weak solutions

I am trying to understand how to solve differential equations of distributions. For example if one consider $ u' + u = \delta_{0}$, where $ u \in \mathcal{D}'$, this would correspond to $<u, ...
0
votes
1answer
57 views

Find limit of a sequence of distributions

I am trying to solve the following exercise: Determine the limit in $\mathcal{D}'(\mathbb{R})$ of $\lim_{t\rightarrow \infty} t^{2}xe^{itx}$, $u_{t} = t^{2}xe^{itx}$. I have tried evaluating the ...
1
vote
0answers
35 views

Calderón-Zygmund $\times$ Schwartz $=$ Calderón-Zygmund

I am in a functional analysis class, and we are being asked to show that if $\eta$ is a Schwartz function and $K$ is a Calderón-Zygmund distribution, then their product is also a Calderón-Zygmund ...
1
vote
0answers
46 views

Convolution of two delta distributions

Show ${\int}_0^{\infty}\delta(x+z)\delta(y-z)dz =\delta(y+x)$ It seems obvious, and I don't think we need a rigorous proof for this (statistical mechanics homework) but I want a rigorous proof of ...
6
votes
4answers
201 views

Dirac delta of nonlinear multivariable arguments

How does one compute a dirac delta function with a multivariable argument? For example, compute: $$ \int^{\infty}_{-\infty}{\rm d}x\,{\rm d}y\, \delta\left(x^{2} + y^{2} - 4\right) ...
2
votes
1answer
40 views

A distribution $u=\frac{1}{x}$

I am interested in finding a distribution $u \in \mathcal{D}'(\mathbb{R})$ such that $u=0$ on $(-\infty,0)$ and $u=\frac{1}{x}$ on $(0,\infty)$. This is exercise 1.4 in Friedlander. Hints or help ...
-1
votes
1answer
42 views

Handling Convergence for Derivative of a Distribution

Obtain the derivative of the distribution defined by $\rho [t] = \int_0^\infty \frac{t(x)}{\sqrt{x}}dx$, and express your answer in the form of an integral over $x$ of a formula that involves $t(x)$ ...
1
vote
1answer
66 views

About a metric over $C^{\infty}(\Omega)$

I need some help with this exercise: let $\Omega$ be an open subset of $\mathbb{R}^n$. We consider: $K_m=\lbrace{x\in\Omega, d(x,\mathbb{R}^n-\Omega)\geq\frac{1}{m},|x|\leq m}\rbrace$ If $\Phi\in ...
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
168 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
334 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 ...
3
votes
1answer
92 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) = ...
1
vote
0answers
24 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 ...
0
votes
1answer
135 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
81 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 $L=L^*$ ...
5
votes
4answers
250 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; $$ \int_{-\infty}^{\infty} g'(x)h(x)\,dx = \int_{-\infty}^{\infty} g(x)h'(x)\,dx.$$ ...
3
votes
1answer
133 views

Convergence of convolution of $L^p$ function with a sequence of distributions

let $h_n\in C_c^\infty (\mathbb{R}^d)$ s.t. $\int h_n dm = 1$ and $\operatorname{supp}(h_n)\to {0}$. I've proven that $h_n\to\delta_0$ in $\mathcal{D}'(\mathbb{R}^d)$, now I'm trying to show that for ...
4
votes
1answer
91 views

How to prove that $x \rightarrow e^{1/x}$ is not a restriction of any real distribution to $ \mathbb {R}_+$?

This is an excercise 2.2 from Hormander, vol. I: Does there exist a distribution $u$ on $\mathbb{R}$ with the restriction $x \rightarrow e^{1/x}$ to $\mathbb{R}_+$? The answer, provided in the book, ...
2
votes
0answers
96 views

Liouville's Theorem in $\mathbb{R}^n$

Liouville's Theorem states that if a tempered distribution is harmonic, $\Delta{u}=0$, then $u$ is given by a polynomial. For the argument, we take Fourier transform of $\Delta{u}=0$ to obtain ...
2
votes
2answers
214 views

Regarding the definition of Schwartz Space of functions

I came across a definition of Schwartz Space where they were defined as functions $f$ such that $\mathrm{lim}_{|x|\to \infty} |x^{\alpha}D^{\beta}f(x)|=0$ for any pair of multiindices $\alpha,\ ...
4
votes
1answer
172 views

A question about convolution of two distributions

Generally,when taking convolution of two distributions,at least one of which is supposed to be of compact support. But when u,$v\in S'(\mathbb{R})$ ( temperate distributions) have suports on the ...
3
votes
1answer
273 views

The distribution $\Delta u$ (where $u = \ln|\vec{x}|$)

Problem Consider the function $u(\vec{x})=\ln|\vec{x}|$ as a distribution on $\mathbb{R}^3$ and $\mathbb{R}^2$. We want to determine $\Delta u$ in the distribution sense. First calculate $\Delta u$ ...
0
votes
1answer
996 views

Fourier transformation of sin, cos, sinh and cosh

I am trying to solve the following exercise Use $\mathcal{F}(e^{xb}) = 2\pi \delta_{ib}$ to calculate the Fourier-Transformation of $\sin x$, $\cos x$, $\sinh x$ and $\cosh x$ Now I am a little ...
1
vote
1answer
68 views

whats the order of a distributional derivate?

I have to calculate the derivatives of order $\le 2$ of for example $f(x) = |x|$, is it the same as the second derivate, what does this "of order $\le 2$" mean? calculating distributionell derivatives ...
1
vote
1answer
38 views

connection between the support and the representation of a distribution

I want to show, that for $u' \in \mathcal{D}'(\mathbb{R}^n)$ supp $u$ = $\{ 0 \}$ iff there exist numbers $m \in \mathbb{N}, c_{\alpha} \in \mathbb{K}$ such that $u = \sum_{|\alpha| \le m} c_{\alpha} ...
1
vote
1answer
263 views

Identify distribution by a constant function [duplicate]

Possible Duplicate: On distributions over $\mathbb R$ whose derivatives vanishes Why can I identify a distribution $G \in \mathcal{D}'((a,b))$, $\partial G = 0$ by a constant function?
2
votes
1answer
505 views

Dirac delta and derivative inside an integral

In development of a calculus (in GR but it doesn't matter here) I have seen one dubious substitution: in an integral of the form: $$\int dx ~\delta(x-x_0)~\partial_x F(x) $$ The author substitutes ...
2
votes
0answers
80 views

Deduce the global differential equation from the pointwisely defined equation in Fourier space

Let $G\in \mathcal{F}(\mathbb{R}^{n+1})'$ be a distribution on the space of spatial Fourier transform'able function, ie an $L^1_{\mathrm{loc}}(\mathbb{R^{n+1}})$ function, $G = G(t,\xi)$. Assume ...