# Tagged Questions

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### 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 ...
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### 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?
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### 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 ...
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### 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, ...
389 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 ...
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### 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 ...
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### 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)?
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 ...
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 ...