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).
2
votes
0answers
22 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
36 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
36 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
39 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
51 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
38 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
28 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
33 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
17 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
52 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
35 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)
$$f(x)=\left\{
\begin{array}{l l l l l}
0, & \quad\text{$ x\leq 0 $}\\
1, ...
0
votes
1answer
55 views
Show that $\delta(\xi-x)=\delta(x-\xi)$
How would you show $\delta(\xi-x)=\delta(x-\xi)$ if you know that
$$\int _{-\infty}^{\infty}\delta(x)h(x)=h(0)$$
Also how would you then show more generally that if $f(\xi)$ is a monotonic ...
1
vote
1answer
36 views
Find $r(x)$ such that $r(x)L$ is self-adjoint
The differential operator
$$L=a(x)\frac{d^2}{dx^2}+b(x)\frac{d}{dx}+c(x)$$
is not self adjoint. How would you find r(x) such that r(x)L is self adjoint.
I know that this is self adjoint when $L=L^*$ ...
3
votes
4answers
82 views
delta functions $e^{x}\delta (x)=\delta (x)$
How would you prove that;
$$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
101 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).
...
