54
votes

### 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 ...

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 ...

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 $f^\beta$ ...

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: ...

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 ...

22
votes

Accepted

### 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 ...

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 ...

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 ...

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 ...

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" ...

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-...

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 ...

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 (...

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 ...

13
votes

### 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....

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 ...

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....

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

Community wiki

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 $...

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^{...

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 (...

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_{...

9
votes

### Eigenvectors of the Fourier transformation

For any even function, $\hat{f}(x) + f(x)$ is a fixed point of the Fourier transform.

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(\...

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