# Tagged Questions

**3**

votes

**1**answer

190 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 ...

**5**

votes

**1**answer

554 views

### Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$.
When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?

**2**

votes

**0**answers

171 views

### Spectral gap of tempered distributions

Hi,
Let $\Lambda\subset\mathbb{R}$ be an infinite discrete set of finite density (for simplicity one may take the density equals 1) and $\delta_{\lambda}$
is a unit mass located at the point ...

**4**

votes

**2**answers

428 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, ...

**2**

votes

**1**answer

386 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**

vote

**1**answer

376 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**

vote

**1**answer

394 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 ...

**1**

vote

**3**answers

613 views

### Solving $x\partial_x f = 0$ over distributions

Solving $x\partial_x f = 0$ over 'normal' functions is the same as solving $\partial_x f = 0$, i.e. one gets $f(x)=c_1$ as the complete answer. But over distributions (if my calculations are ...

**2**

votes

**5**answers

2k views

### fourier transform of (real) exponential

Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?

**1**

vote

**2**answers

409 views

### What do we know about the space of finite order distributions ?

Hi,
(Question updated)
My question is about the space of distributions of finite order $\mathcal{D}'_F$ (say on $\mathbb{R}^n$). What do we know about it ?
From in the information I gathered, it ...

**23**

votes

**7**answers

2k views

### What examples of distributions should I keep in mind?

I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition?
Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write ...