54
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A roadmap to Hairer's theory for taming infinities
Let me try to expand a little bit on Ofer's answer, in particular on points 1-3.
These functions (or rather distributions in general) are essentially the multilinear functionals of the driving noise ...
- 8,464
30
votes
A roadmap to Hairer's theory for taming infinities
Let me comment on points 4) and 5). The problem with infinities in QFT or traditional equilibrium statistical field theory
is related to the one addressed by Martin's theory but there are some ...
- 21.4k
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 $f^\beta$ ...
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27
votes
Accepted
Nice applications for Schwartz distributions
Unsurprisingly, the topics that occur to me have various connections to number theory (and related harmonic analysis) (and unclear to me what might have already been done in your course...):
EDIT: ...
- 22.3k
23
votes
Eigenvectors of the Fourier transformation
Given a square-integrable, positive semi-definite function $f$, with its Fourier transform $\hat{f}$, then the function
$$F=f^2+\hat{f}\star\hat{f},$$
with $\star$ the convolution, is its own ...
- 172k
22
votes
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Anti-delta function?
There's quite a bit on this in Harold Jeffreys's unusual book Theory of Probability, in which such a function is one of the more prominent examples of “improper priors.” In Robert, Chopin, and ...
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21
votes
A roadmap to Hairer's theory for taming infinities
There are several treatments of Hairer's theory, including lecture notes of his that try to give the "big picture".
Brief answers to some of your questions:
1) Those are the solutions to the ...
- 7,274
21
votes
How to generalize the various vector calculus theorems to distributions?
This may be borderline for this site, but here goes: What you are looking for may be the notion of weak derivative. A function $f$ defined on some $d$-dimensional set $\Omega$ has a weak partial ...
- 12.2k
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 ...
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17
votes
Can distribution theory be developed Riemann-free?
I will only consider the temperate situation (involving the spaces $\mathscr{S}$, $\mathscr{S}'$ and $\mathscr{O}_{\rm M}$) and I will only discuss the first example as well as a "dual" ...
- 21.4k
16
votes
Accepted
Why is multiplication on the space of smooth functions with compact support continuous?
You can spare yourself the functional analytic abstract nonsense by using an explicit set of seminorms on $\mathcal{D}(\mathbb{R}^d)=C_{c}^{\infty}(\mathbb{R}^d)$
which, unfortunately, are not well-...
- 21.4k
15
votes
Accepted
Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?
No, the most obvious example is the reflection operator: $Rf(x) = f(-x)$ this is not pseudolocal (in fact the $\xi$-compotent of the wavefront set gets a sign flip). Also the Fourier transform.
More ...
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15
votes
Early successes of Schwartz distribution theory
Following the citation for the 1950 Fields medal I would argue that putting the Dirac delta function on a firm ground was the early success of the theory of distributions.$^\ast$
An extensive list of (...
- 172k
14
votes
The "Spaces of Schwartz distributions are finite dimensional" challenge
Like others have pointed out the key, concept is that of nuclear spaces and a good presentation can be found in volume 4 of Gelfand's Generalized Functions.
Kolmogorov has introduced a concept ...
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13
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The "Spaces of Schwartz distributions are finite dimensional" challenge
Every short exact sequence $$0\to \mathscr S'\to Y\to \mathscr S'\to 0$$ (where $Y$ is any locally convex space) splits, i.e., the quotient map $Y\to \mathscr S'$ has a continuous linear right inverse....
- 15.3k
13
votes
How to generalize the various vector calculus theorems to distributions?
This is an addition to Dirk's answer. It is of course impossible to give an all-embracing answer to your question—there are so many identities. But the general answer is a resounding YES. The ...
- 266
13
votes
Accepted
Why are distributions "tempered"?
Can someone explain, why in English the name "tempered" wins?
Presumably because that’s how the inventor himself translated it (French past participle to English past participle), on e.g. p....
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12
votes
Poincaré lemma for distributions
See page 20 of J P Demailly Complex Analytic geometry.The other source to look at is de Rham's book on differentiable manifolds.Yet another source to look at is Laurent Schwartz Theorie des ...
- 4,101
12
votes
Nice applications for Schwartz distributions
Two (edit: now four) not-so-usual examples come to my mind:
There is the proof of the central limit theorem using Fourier analysis, as done in Chapter 7 of Hörmander's book. It's a cornerstone of ...
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12
votes
Accepted
Representation of the Dirac delta function
As usual in such examples, there is no need to integrate against a test function. One can simply use the fact that if a sequence (or net) of distributions converges in the distributional sense, then ...
- 206
11
votes
Are Fourier transforms of L^p stable under diffeomorphisms?
Don't think so. Consider the surface measure on a compact surface (e.g. a sphere). Its Fourier transform has a rate of decay which depends on the curvature of the surface; more precisely, on the order ...
- 8,621
11
votes
Accepted
Chain rule in Sobolev space
The main difficulty in the proof of the rule is to prove that $\nabla u=0$ a.e. on the set $u^{-1}(\Sigma)$, where $\Sigma$ is the set where $F$ is not differentiable; and where $\nabla u=0$ one ...
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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 ...
10
votes
Accepted
Prove that a given distribution is tempered
In Laurent Schwartz's Théorie des Distributions (page 245, chap. VII, §5) you can find something similar: A distribution $T\in \mathscr D'(\mathbb R^d)$ is tempered if and only if all regularizations $...
- 15.3k
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^{...
- 172k
10
votes
Accepted
Non-Schwartz test functions for the explicit formula for L-functions
The class of "weights" $\phi(t)$ for which the explicit formula holds is pretty flexible. The particular choice of weight depends on the problem one studies. On one hand, sums over primes (...
- 5,014
9
votes
Accepted
Nice way to express $H^{-1}(\mathbb{S}^1)$
I am somewhat confused that, despite saying many true and useful things, no one has said directly that $H^{-1}$ on the circle can be characterized as the set of distributions $\theta$ such that $\sum_{...
- 22.3k
9
votes
Eigenvectors of the Fourier transformation
For any even function, $\hat{f}(x) + f(x)$ is a fixed point of the Fourier transform.
- 95.2k
9
votes
The "Spaces of Schwartz distributions are finite dimensional" challenge
The Schwartz kernel theorem: For a manifold $M$ we have
\begin{multline*}
\mathcal D'(M\times M) =
(\mathcal D(M)\hat{\hat\otimes} \mathcal D(M))'
= (\mathcal D(M)\hat\otimes \mathcal D(M))' = L(\...
- 24.8k
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