28
votes
Anti-delta function?
Let $\beta\mathbb{R}$ be the Stone-Čech compactification of $\mathbb{R}$: any bounded continuous function $f$ on $\mathbb{R}$ extends uniquely to a (necessarily bounded) continuous function $f^\beta$ ...
- 32.4k
18
votes
Accepted
Research topics in distribution theory
While I do not know much about current development of the general theory of distributions, I can say something about the current research topics in a special class of distributions, the theory of ...
- 28k
10
votes
Anti-delta function?
To elaborate on the comment, I would suggest to take $F(x)=x^{-2}\delta(1/x)$. Let me check for the representation
$\delta_{\epsilon}(x)=(2\pi\epsilon)^{-1/2}e^{-x^2/2\epsilon}$,
and $F_\epsilon(x)=x^{...
- 188k
10
votes
Research topics in distribution theory
I think this is a good question. I am no longer working in this field. When I was working in this field, there is no general program and I do not know what are the important open problems. Instead of ...
9
votes
Anti-delta function?
This is more a long comment following the ones of Mateusz Kwasnicki and Anixx and the nice answers given by Michael Hardy, Gro-Tsen, et al.
As stated by Mateusz, such a mathematical objiect cannot be ...
- 6,357
7
votes
Research topics in distribution theory
I Don't have time for a very elaborate answer (will expand later), but I think the main research questions about Schwartz distribution relate to probability theory on spaces of such distributions. ...
6
votes
Generalized functions in infinite dimensions
The study of an "infinite-dimensional delta function" has been motivated to a large extent by applications in quantum field theory. Here is some relevant literature:
Tempered distributions ...
- 188k
5
votes
Anti-delta function?
It seems to me that, insofar as a "generalized distribution" like your $F$ exists in some sense, there would have to be a continuum of them, corresponding to different possible values $0 ≤ c ...
- 1,289
4
votes
Anti-delta function?
We can define your "anti-delta function" as an unusual type of distribution.
First, describe $F$ in terms of the integral $\langle F,\varphi\rangle:=\int_{-\infty}^{\infty}F(x)\,\varphi(x)\,...
- 141
4
votes
Colombeau generalized functions
This recent textbook might be helpful:
Geometric Theory of Generalized Functions with Applications to General Relativity, M. Grosser, M. Kunzinger, M. Oberguggenberger, and R. Steinbauer (2013).
...
- 188k
3
votes
Accepted
English translation of Schwartz's papers on vector-valued distributions
These papers have not been translated, as far as I know, however there exist lecture notes in english of courses by Schwarz on this topic:
• Introduction to the Theory of Distributions
• Lectures on ...
- 188k
3
votes
Accepted
What is the distribution of the following limit?
This question already has an impeccable answer but I would like to give a second one in the hope that it might be of interest to the OP. Before so doing, I will list the properties of distributions (...
- 266
2
votes
Accepted
Is there a name for this nice property of the usual weak incompressible Navier-Stokes equation?
A smooth function that satisfies the weak formulation is also a strong solution is called equivalent and a strong solution that satisfies the weak formulation is consistent.
- 144
2
votes
Accepted
Description of $\mathcal{S}^{2}_0(\mathbb{R})$ and certain class of functions inside it
$\newcommand\S{\mathcal S}\newcommand\R{\mathbb R}$For $x\in(0,1)$, let $g(x):=\frac1x$, and then let
$$f(x):=e^{-g(x)-g(1-x)}$$
for $x\in(0,1)$, with $f(x):=0$ for real $x\notin(0,1)$.
Then $f\in\S_0^...
- 128k
2
votes
Accepted
Interchanging Integration Order involving Fourier Transform
The subject of definite integrals for distributions was investigated in some detail by several mathematicians in the 50’s—-they used an elementary approach based on the fact that distributions are (...
- 2,472
2
votes
Accepted
Schwartz distributions, Colombeau algebra and applications
For the basics (= Chapter 1) you only need (functional) analysis and distribution theory. Then you also need infinite dimensional analysis (convenient calculus) but that is also contained in Chapter 2....
- 659
2
votes
What is the distribution of the following limit?
$\newcommand{\ep}{\epsilon}\newcommand{\de}{\delta}\newcommand{\R}{\mathbb{R}}$Let
\begin{equation*}
K_\ep(x):=\frac{\ln^m(x-i\ep)}{x-i\ep}-\frac{\ln^m(x+i\ep)}{x+i\ep}.
\end{equation*}
Let us ...
- 128k
2
votes
Accepted
Why does the formula (7) in the article equivariant cohomology with generalized coefficients hold
I guess the point is that $$ \int_{\mathfrak g} \alpha(X) \Phi(X) dX ,$$ by definition, is $$(\alpha, \Phi)$$ since the idea of generalized functions is that the linear form represents the integral. ...
- 148k
1
vote
Distribution boundary value of analytic function and wave front sets
You need to check Condition (ii) in Definition 8.2.2 in the first volume of Hörmander's ALPDO. Let us note $f(x+i0)$ the limit-distribution of your question and let $\Gamma$ be its wave-front-set. Let ...
- 16.2k
1
vote
When can we integrate a distribution over a smooth domain?
A simple and elementary theory of integrals of distributions was developed by several authors in the $50$'s and $60$'s. Since every distribuion has a primitive, this reduces to defining the limit of ...
- 19
1
vote
the relation between a continuous family of distributions and a distribution of 2 variables
We have describe the constriction and its relation with WF and Hormander's restriction in section 2.3 of https://arxiv.org/pdf/1212.3630.pdf
- 2,639
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