All Questions
13 questions
3
votes
1
answer
136
views
Functions of moderate increase compactly generated?
Let $\mathcal{O}_M(\mathbb{R}^d)$ be the space of smooth moderately increasing functions
$\{ f \in \mathcal{C}^\infty(\mathbb{R}^d) : \forall \alpha \exists N \text{ such that} \Vert \langle \cdot \...
- 131
3
votes
1
answer
264
views
How to prove that this one-parameter family of distributions converges to the Dirac measure?
While trying to understand a proof in a paper, I came upon the following a calculation needing the following identity:
$$\lim_{t\to 0} \int_{-\infty}^\infty \left(e^{-\log(4\pi i t)/2} e^{ik^2/4t} -\...
- 840
7
votes
1
answer
2k
views
Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$
I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...
- 1,247
2
votes
0
answers
89
views
Extension of a generalized function to the plane
Let $\phi$ be a generalized function on $\mathbb{R}^2\backslash\{0\}$, and assume that its differential $d\phi$ extends to the whole plane $\mathbb{R}^2$.
Q. Does $\phi$ also extend to $\mathbb{R}^...
- 21.8k
5
votes
2
answers
622
views
Completeness of an exponential family
The question is this: Does there exist an integrable function $f\colon\mathbb R\to\mathbb R$ such that $f$ differs from $0$ on a set of nonzero Lebesgue measure and
\begin{equation}
\int_{\mathbb R}...
- 128k
15
votes
2
answers
680
views
Are Fourier transforms of L^p stable under diffeomorphisms?
Let $\xi$ be a compactly supported distribution on $\mathbb R^n$ and assume that its Fourier transform is in $L^p$. Let $\phi:\mathbb R^n\to\mathbb R^n$ be a diffeomorphism. Does the Fourier ...
- 2,639
3
votes
2
answers
279
views
Nice way to express $H^{-1}(\mathbb{S}^1)$
I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
- 33
3
votes
1
answer
916
views
Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?
Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by
\begin{equation*}
X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}.
\end{equation*}
My ...
- 135
4
votes
2
answers
974
views
Wightman fields vs local functionals vs operators
In QFT literature one wants to look at $n-$point correlation functions of "operators" inserted at $x$ say, $\cal{O}(x)$ and if $\phi_i$ are the fields then the quantity one has in mind is written as, $...
- 3,541
2
votes
1
answer
535
views
about decomposition of a non-negative definite operators
Hello,
Many years before, I had the following problem.
We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called separable if it ...
- 1,649
2
votes
1
answer
547
views
Equivalent references for Schwartz's book of the distribution theory
Hello,
It seems that there is no English translation of the Schwartz's book 1966. I may need to use the spaces like
$$
\dot{\mathcal{B}}(R),\quad \dot{\mathcal{B}}'(R),\quad \mathcal{B}(R),\quad \...
- 1,649
1
vote
0
answers
693
views
A question about an equivalent definition of the Schwartz distribution
Hello,
Does anyone know a reference or proof of the "if" part of the following statement?
$$
\mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\...
- 1,649
25
votes
6
answers
15k
views
Does every distribution define a Radon measure?
On the one hand, Wikipedia suggests that every distribution defines a Radon measure:
http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions (revision from February 2010, ...
- 2,895